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2009, 113, 11976–11979 Published on Web 06/19/2009
New Insights on the Growth of Anisotropic Nanoparticles from Total Energy Calculations Tu´lio C. R. Rocha,†,‡,| Fernando Sato,‡,§ So´crates O. Dantas,§ Douglas S. Galva˜o,‡ and Daniela Zanchet*,† Laborato´rio Nacional de Luz Sı´ncrotron -LNLS, CP 6192, CEP 13083-970, Campinas -SP, Brazil, Instituto de Fı´sica Gleb Wataghin, UniVersidade Estadual de Campinas, CP 6165, CEP 13083-970, Campinas -SP, Brazil, and Departamento de Fı´sica, ICE, UniVersidade Federal de Juiz de Fora, CEP 36036-330 Juiz de Fora -MG, Brazil ReceiVed: April 24, 2009; ReVised Manuscript ReceiVed: June 7, 2009
The growth mechanism of anisotropic metallic nanoparticles is still an open and polemical question. The common observation of the existence of nonspherical (not the most stable) shapes in varied experimental conditions is not fully understood. In this work, based on results from total energy calculations for different shapes and sizes of Ag nanoparticles, we provide new insights of why anisotropic structures are commonly found in different preparation conditions. We show that, assuming the presence of a particle shape distribution in the beginning of the growth process, anisotropic nanoparticles can preferentially grow over spherical ones due to the fact that the energy required to build larger anisotropic structures could be less than the one required to build isotropic structures. These results suggest that many previous works in literature shall be revisited accordingly to these new finds. Metal nanoparticles with anisotropic shapes have attracted great interest because of their potential importance in different technological applications, such as catalysis,1 biological labeling,2 subwavelengths optics,3 and surface-enhanced Raman scattering.4 They are also very interesting structures from pure scientific point of view with many unsolved issues, such as the origin of the anisotropy, the shape control in the 1-5 nm size range and more generally the materials’ properties dependency on shape (surface plasmon excitation, magnetism, superconductivity). Wet chemical synthesis has been demonstrated to be a promising method to produce these particles with high yields, tunability of particle size, shape, structure, composition, and surface chemistry, also with potential for high-volume production. Many examples of silver and gold anisotropic colloids can be found in the literature, such as rods, wires, disks, triangular prisms, cubes, and branched particles.5 However, the formation of these morphologies has not been fully understood yet and a general mechanism is still a challenge. Common arguments in many works are that capping molecules preferentially bound to specific facets can block/enhance the growth in some directions6 and the formation of surfactant micelles would physically direct the growth.7 More recently, some works have pointed out structural defects as key elements directing the anisotropic growth.8-10 Other aspects such as oxidative etching and control of the nucleation have also been proposed.11,12 Recent comprehensive reviews have been published by Xia et al.13 and Pastoriza-Santos and Liz-Marzan.14 However, a fundamental question remains to be addressed, why are aniso* To whom correspondence should be addressed. Tel.: +55 19 35121010.Fax: +55 19 3512-1004. E-mail:
[email protected]. † Laborato´rio Nacional de Luz Sı´ncrotron -LNLS. ‡ Universidade Estadual de Campinas. § Universidade Federal de Juiz de Fora. | Present address: Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany.
10.1021/jp903794y CCC: $40.75
tropic structures (less stable in comparison with spherical ones) so commonly found? In this work, we investigated the energetic balance of particles with different shapes and sizes in order to evaluate the importance of these aspects in the preferential (and sometimes much faster) growth of anisotropic structures. The realistic description of the energetics of transition metal nanostructures with thousands of atoms is still challenging. The use of sophisticated ab initio methods is not feasible due to the extremely high computational cost and other methods must be used. One of the few available methodologies capable of reliably calculating structural properties of metallic structures containing ∼100 000 atoms is tight-binding models with second-moment approximations,15-21 and it was our choice in the present work. The second moment approximation although not an explicit full quantum method (the size of the structures investigated here precludes the use of such methods) goes beyond geometrical features and incorporates quantum features through parametrization. It is based on the well-known fact that cohesive properties of transition metal and their alloys originate mainly from the large d-band density of states (DOS). The methodology is completely general and can be used for study face-centered cubic (fcc) and body-centered cubic (bcc) structures. It contains two major terms, the first one is the repulsive contribution, normally assumed as pairwise and described by Born-Mayer type interaction and the second term ensures system stability and is an attractive interaction (SMA) quantum mechanical in origin that incorporates many-body summations. For details see ref 20. The structural models of silver nanospheres, nanodisks, nanorods, and triangular nanoprisms (the most common shapes found in the experiments) were generated by truncation of a fcc lattice with number of atoms varying from 2000 up to 82 000 atoms (nearly 4 to 14 nm in size) (table 1). Illustrative pictures of the structural models are shown in Figure 1. The sizes and low aspect ratio of the nanoparticles were chosen based on 2009 American Chemical Society
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TABLE 1: Size, Number of Atoms, and Calculated Energies Per Atom (eV/atom-1) for Each Structural Model (for Anisotropic Particles Dimensions Are Given As Largest Axis × Smallest Axis) size in nm (number of atoms) (energies per atom (eV /atom-1))
shapea sphere disk rod prism a
4.5 (2006) (-2.836) 4.0 × 3.1 (2257) (-2.834) 5.9 × 3.2 (2035) (-2.820) 5.2 × 3.3 (2030) (-2.810)
6.4 (7854) (-2.883) 5.8 × 4.5 (6789) (-2.873) 8.5 × 4.5 (7760) (-2.872) 7.0 × 4.5 (7144) (-2.861)
10.1 (28918) (-2.910) 9.5 × 7.3 (28677) (-2.906) 13.4 × 7.3 (29034) (-2.902) 11.4 × 7.3 (29910) (-2.998)
13.8 (81856) (-2.924) 13.2 × 10.1 (82179) (-2.922) 19.1 × 10.1 (82080) (-2.919) 15.8 × 10.1 (82260) (-2.936)
Despite not having been included in our calculations, hexagonal nanoparticles are expected to be very similar to prisms.
quantities that scales with the area (surface energy, planar defects, etc.), the third one is related to factors that scales with lengths (linear defects, edges, etc.), and finally a last general term, independent of N (a, b, c, and d being constants). N is the total number of atoms of the structure. The coefficients that characterize this energy function for each nanoparticle should vary as shape, surface, and structure are changed; being the magnitude of the variations determined by electronic characteristics of the material. Defining the constant term, d, to be zero as our energy reference, the energy per atom (ε) as function of the N-1/3 can be described as
ε(u) ) A + Bu + Cu2
Figure 1. Structural models used to calculate the configuration energies. (a) Sphere; (b) triangular nanoprism; (c) disk and; (d) rod.
experimental results, as in the case of seed-mediated photochemical growth of Ag nanoprisms.22,23 The same aspect ratio was used for all sizes for comparison purposes. These models were then subjected to geometrically constrained optimizations in molecular dynamics calculations. The steps of molecular dynamics were made to address temperatures near 0 K and kinetic energies close to 1.0 × 10-10 eV. This approach allowed us to optimize all distances among atoms preserving their initial topological features, through constraining the structures in their local minima. It should be stressed that a full relaxation without constraints is not feasible for anisotropic particles, since they would evolve to structures with smaller surface area and lower energies. Finally, the energy per atom for each optimized model and different sizes was obtained. In the present work, we used the same parameters from ref 16. The results are shown in Table 1 with the data about sizes and number of atoms of each model. It is expected that aspects such as cohesion, surface, defects, edges, strain, and total number of atoms, among others, will contribute differently (in relative terms) to the total energy of the nanoparticles of different sizes and/or shapes. In order to better evaluate these contributions as a function of size and topology, we can generalize the functional dependence of the total energy as
E(N) ) aN + bN2/3 + cN1/3 + d
(1)
where the first term represents quantities that scales with the volume (cohesion, strain, etc.), the second one describes the
(2)
where u ) N-1/3. Figure 2 summarizes the energetic aspects associated with the isotropic and anisotropic structures. Figure 2a shows the calculated energies per atom, presented in table 1, as a function of u. As expected, the spherical nanoparticles always have smaller energies. The calculated energy values exhibit a linear behavior, indicating that the total energy is dominated by terms that scale with area, in our case, the surface energy. Linear fittings were then carried out to extrapolate the values to particles with large number of atoms (small u values). The results pointed out nearly the same value for the parameter A (the term that scales with volume) for all of the morphologies; the B value was smaller for the spheres and larger for the prisms. The linear behavior also means that the quadratic terms in u, that is, energy terms that scale with length have negligible contribution in the analyzed size range. In other words, similar A values and different B values (as well as negligible C values) mean that in the size range considered in this work and within the approximation of our calculations, the dominant terms that differentiate the total energy of the particles as a function of shape scales with the area. This is somehow expected considering that we are dealing with particles with thousands of atoms. As we can see from Table 1, the stability decreases from sphere, disk, rod, and prism, for all sizes. Looking more carefully to the angular coefficients in Figure 2a, the results pointed out that the decrease in total energy is larger for anisotropic nanoparticles (large B values) than for the spherical ones. This is better shown in Figure 2b where the differences in energy per atom (the energetic gain, ∆E) for each shape are shown as a function of the size increase. Figure 2b was generated by the interpolation of the calculated energy per atom values for particles containing ∼2500 up to ∼80 000 atoms (doubling the number of atoms in each step). It is clear from Figure 2 that although the sphere is always the most stable morphology for all sizes, the decrease in the energy per atom for particles with different sizes is larger for
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Figure 2. (a) Calculated energy per atom as function of u ) N-1/3. The coefficients of the linear fittings are indicated as inset in the figure. (b) Semilog plot of the difference in energy per atom (energetic gain, ∆E), as function of the increase in the number of atoms (∆N) for nanoparticles with different shapes.
Figure 3. Schematic view of the energetic path of the nanoprism growth compared to the sphere from an energetic point of view. The energy levels of particles with different shape were connected by a high energy barrier to represent that the particles are not allowed to change shape in our calculations (see text for detail).
anisotropic particles. This energetic gain (∆E) is higher for the anisotropic particles relative to spheres at smaller sizes and decreases as the size is increased (but always higher). This implies that one needs less energy to grow a larger prism from a smaller one than to do the same for the spheres. This analysis indicates that the growth of anisotropic particles is energetically favored in comparison with spherical particles. This is a direct consequence of the energetics being dominated by surface energy in this size range, as previously observed in the energy curves. Anisotropic nanoparticles have higher surface to volume ratios than spheres and, consequently, higher energy per atom, even though they can expose facets with lower surface energies (111 in fcc metals, for example). Moreover, the surface to volume ratio scales inversely with the size of the particles. Hence, whenever anisotropic nanoparticles (more energetic) are formed together with spherical ones (less energetic) and their shape is not allowed to change, the reduction in the total surface to volume ratio of the system is higher and consequently in the total energy per atom, when the anisotropic particles increase in size. Figure 3 shows a schematic view of the process. If we assume the presence of a particle shape distribution in the beginning of the growth process, it costs less energy to increase the size of the anisotropic particles than to increase the size of the spheres. In other words, if anisotropic nanoparticles are somehow formed (and cannot change shape), the energetic gain in their growth is larger than the growth of spheres with similar sizes, as shown in Figure 2. The energy levels of particles with different shape are schematically connected by a high energy barrier that avoids shape change.
For colloidal nanoparticles in solution at room temperature or in the presence of capping agents, it is reasonable to expect that an energy barrier to change shapes exist, keeping the particle growth in the same path (see Figure 3). This energy barrier has also been suggested in the growth of CdSe24 and Pt25 anisotropic nanoparticles. It has been reported in the literature26 that the temperature or the aging of the colloidal solution in some cases can lead shape transformations, which in our scheme would be represented by crossing the energy barrier, but this issue was not investigated in this work. It is important to remark that this energetic argument is restricted to the growth of anisotropic nanoparticles compared to spheres; it cannot be applied to their formation. The higher energy cost to form anisotropic nanoparticles is already paid in the seed formation. In order to produce anisotropic nanoparticles, the high symmetry of the fcc lattice must be broken, which can be achieved for example by the preferential stabilization of facets, presence of defects, or physical constriction of the growth. One experimental fact that could be explained by this energetic argument is the frequent observation of some large anisotropic particles as a byproduct in chemical synthesis of Au and Ag spherical nanoparticles.27 Another example in which this energetic argument may play an important role is the case of the photochemical-induced growth of triangular nanoprism,10,22,23 which was in fact the motivation for the present work. For this system, small spherical nanoparticles (seeds) in aqueous solution are irradiated by visible light in the presence of additional silver ions and grow forming triangular nanoprisms and also large spherical nanoparticles. In previous works,10,22 some of us have identified two important factors in the formation and growth of these nanoprisms: structural defects and surface plasmon excitations. In the initial stages of the reaction, small nanoprisms are formed due to the anisotropic deposition of Ag atoms over seeds with lamellar defects present in the initial solution, where others seeds grow isotropically. Then, these small nanoprisms grow together with the spheres, until their surface plasmon absorption band matches the incident radiation. From this moment, surface plasmon excitation takes place enhancing the photochemical reaction and the nanoprisms grow faster, reaching their final sizes determined by the wavelength of the incident light. The energetics might act in this system, as an additional driving force in the intermediate stages, being responsible for the preferential growth of the low aspect ratio nanoprisms relative to the spheres just after they have been formed and still not yet absorbing significantly the incident light. The size distribution of the initial spheres and the nanoprisms observed in intermediate stage are in the same size range as the models
Letters used in our calculations. Although kinetic and specific chemical reactivity arguments (such as the important role played by surfactants; see, for example, ref 13) can always be invoked to explain the anisotropic growth, our calculations showed that energetic aspects will also contribute in the same direction. These aspects have been overlooked in these cited examples as well as in other systems. In summary, in this work we addressed the anisotropic growth of Ag nanoprisms from the energetic point of view. By calculating the energy per atom of structural models of different sizes and shapes, we showed that the growth of the Ag nanoprisms is energetically favored once they are formed. It is important to remark that many mechanisms are in fact involved in the complex phenomenon of the growth of anisotropic metal nanoparticles in particular for colloidal nanoparticles. Among them we can mention kinetics aspects, rate of precursor reduction, capping molecules, structural defects, etc. The main objective of the present work was to demonstrate that the energetic aspects can also play a very important role in determining the preferential shape of nanoparticles and these aspects have not been considered in the literature. This work presents a general argument that can explain the anisotropic growth in the case of fcc metallic nanoparticles, presenting the problem from another perspective that may be extended to a broader set of systems. Acknowledgment. The authors thank the Brazilian agencies FAPESP, CAPES, and CNPq, for partial financial support. References and Notes (1) Rioux, R. M.; Song, H.; Grass, M.; Habas, S.; Niesz, K.; Hoefelmeyer, J. D.; Yang, P.; Somorjai, G. A. Top. Cat. 2006, 39, 167. (2) Nam, J. M.; Park, S. J.; Mirkin, C. A. J. Am. Chem. Soc. 2002, 124, 3820. (3) Maier, S. A.; Kik, P. G.; Atwater, H. A.; Meltzer, S.; Harel, E.; Koel, B. E.; Requicha, A. A. G. Nat. Mater. 2003, 2, 229.
J. Phys. Chem. C, Vol. 113, No. 28, 2009 11979 (4) Nie, S.; Emory, S. R. Science. 1997, 275, 1102. (5) See AdV. Mater 2005, 30, special issue on synthesis and plasmonic properties of nanostructure. (6) Caswell, K. K.; Bender, C. M.; Murphy, C. J. Nano Lett. 2003, 3, 667. (7) Pileni, M. P. Nat. Mater. 2003, 2, 145. (8) Lofton, C.; Sigmund, W. AdV. Func. Mater. 2005, 15. (9) Elechiguerra, J. L.; Reyes-Gasga, J.; Yacaman, M. J. J. Mater. Chem. 2006, 16, 3906. (10) Rocha, T. C. R.; Zanchet, D. J. Phys. Chem. C 2007, 111, 6989. (11) Wiley, B.; Herricks, T.; Sun, Y.; Xia, Y. Nano Lett. 2004, 4, 1733. (12) Zhang, W.; Chen, P.; Gao, Q.; Zhang, Y.; Tang, Y. Chem. Mater. 2008, 20, 1699. (13) Xia, Y.; Xiong, Y.; Lim, B.; Skrabalak, S. E. Angew. Chem. Int. Ed. 2009, 48, 60. (14) Pastoriza-Santos, I.; Liz-Marzan, L. M. J. Mater. Chem. 2008, 18, 1724. (15) Coura, P. Z.; Legoas, S. B.; Moreira, A. S.; Sato, F.; Rodrigues, V.; Dantas, S. O.; Ugarte, D.; Galvao, D. S. Nano Lett. 2004, 4, 1187. (16) Cleri, F.; Rosato, V. Phys. ReV. B 1993, 48, 22. (17) Shimizu, F.; Kimizuka, H.; Li, J.; Yip, S. Proceedings of the Fourth International Conference on Supercomputing in Nuclear Applications SNA 2000, 4-7 September, 2000, Tokyo, Japan. (18) Gonzalez, J. C.; Rodrigues, V.; Bettini, J.; Rego, L. G. C.; Rocha, A. R.; Coura, P. Z.; Dantas, S. O.; Sato, F.; Galvao, D. S.; Ugarte, D. Phys. ReV. Lett. 2004, 93, 126103. (19) Bettini, J.; Sato, F.; Coura, P. Z.; Dantas, S. O.; Galva˜o, D. S.; Ugarte, D. Nat. Nanotechnol. 2006, 1, 182. (20) Sato, F.; Moreira, A. S.; Coura, P. Z.; Dantas, S. O.; Legoas, S. B.; Ugarte, D.; Galvao, D. S. Appl. Phys. A: Mater. Sci. Process. 2005, 81, 1527. (21) Rosato, V.; Guillope, M.; Legrand, B. Philos. Mag. A 1989, 59, 321. (22) Rocha, T. C. R.; Winnischofer, H.; Westphal, E.; Zanchet, D. J. Phys. Chem. C 2007, 111, 2885. (23) Maillard, M.; Huang, P.; Brus, L. Nano Lett. 2003, 3, 1611. (24) Lee, S. M.; Cho, S. N.; Cheon, J. AdV. Mater. 2003, 15, 441. (25) Ren, J.; Tilley, R. D. J. Am. Chem. Soc. 2007, 129, 3287. (26) Tang, B.; An, J.; Zheng, X.; Xu, S.; Li, D.; Zhou, J.; Zhao, B.; Xu, W. J. Phys. Chem. C 2008, 112, 18361. (27) Kimling, J.; Maier, M.; Okenve, B.; Kotaidis, V.; Ballot, H.; Plech, A. J. Phys. Chem. B 2006, 110, 15700.
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