NANO LETTERS
Molecular Dynamics Simulation of the Aggregation of Titanium Dioxide Nanocrystals: Preferential Alignment
2009 Vol. 9, No. 12 4198-4203
Mozhgan Alimohammadi† and Kristen A. Fichthorn*,†,‡ Department of Chemical Engineering, Department of Physics, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 Received July 27, 2009; Revised Manuscript Received August 19, 2009
ABSTRACT We use classical molecular-dynamics simulations to study the aggregation of various titanium dioxide (anatase) nanocrystals in vacuum. In all cases, we observe a strong tendency for the nanocrystals to aggregate with certain preferred orientations in a “hinge” mechanism. Although some of the nanocrystals possess significant dipole moments, dipole-dipole interactions do not direct aggregation, implying that higherorder multipole moments are the driving force for preferential alignment. These high-order multipole moments originate from under-coordinated O and Ti surface atoms on the edges between nanocrystal facets, which create localized regions of positive and negative charge. The observed mechanism for preferential alignment may be a driving force for oriented attachment and the growth of anisotropic structures during crystallization.
The aggregation of nanosize clusters occurs in materials synthesis through both colloidal and vapor-phase routes and, thus, plays a role in determining structure. In many cases, nanoparticles aggregate randomly, to produce irregular or fractal-like clusters. However, nanocrystals can also approach one another and merge along specific crystallographic directions to form twinned or single-crystal structures, in the oriented attachment mechanism.1-4 The ability to direct crystallization through oriented attachment is an exciting prospect that could allow for the creation of new nanostructures with well-defined sizes and shapes. To this end, insight into its origins, which are not currently clear in all cases, would be beneficial. Atomic-scale investigations could contribute significantly to understanding oriented attachment. Although understanding the forces and dynamics between nanometer-sized particles in the presence of solvent (and perhaps additives) is currently challenging for first-principles simulations, classical molecular dynamics (MD) can simulate nanoparticle aggregation5-11 and furnish many details of this process that are not accessible experimentally. In this work, we focus on the sintering of nanocrystalline titanium dioxide (anatase) in vacuum. Oriented attachment was discovered in experimental studies of the crystallization of colloidal anatase.3 In a colloidal system, oriented attachment could originate from intrinsic forces between nanoparticles, such as dipole-dipole interactions,1,2,12-14 or from selective adsorption and surface chemistry of liquid-phase * Towhomcorrespondenceshouldbeaddressed.E-mail:
[email protected]. † Department of Chemical Engineering. ‡ Department of Physics. 10.1021/nl9024215 CCC: $40.75 Published on Web 08/31/2009
2009 American Chemical Society
molecules at the nanoparticle-liquid interface.1,2,15 Our recent MD simulations indicate that oriented attachment could generally arise through interactions mediated by the solvent.16,17 It is also possible that a synergistic interplay between these various phenomena could lead to a directional preference for aggregation.1,2,12,13,18,19 By studying the aggregation of nanoparticles in vacuum, we isolate the role of intrinsic forces between the nanoparticles. We consider charge-neutral anatase nanocrystals with variations of the truncated tetragonal bipyramidal Wulff shape predicted by first-principles calculations based on density-functional theory (DFT).20,21 Nanocrystalline anatase has been experimentally observed to possess approximate Wulff shapes.3,22,23 As shown in Figure 1a,b, nanocrystals with symmetric Wulff shapes contain both {101} and {001} facets. We consider Wulff-shaped nanoparticles of two different sizes, large nanoparticles that consist of 3774 Ti and O atoms (Figure 1a) and small nanoparticles containing 816 atoms (Figure 1b). The length-to-width ratios of these nanocrystals were selected to approximately match those predicted in the DFT study by Barnard and Zapol.21 In addition to Wulff shapes, we considered three asymmetric nanocrystals, shown in Figure 1c-e, which mimic possible off-Wulff shapes that could occur during crystal growth. These asymmetric nanoparticles possess permanent dipole moments, while the symmetric nanocrystals do not. Thus, we can study the effect of dipole-dipole interactions, which have been proposed to play a role in oriented attachment.1,2,12-14 In the asymmetric nanoparticle shown in Figure 1c, the general Wulff shape is maintained but one of
Figure 1. Equilibrated structures of the large (a) and small (b) symmetric nanoparticles with Wulff shapes, a large nanoparticle that is truncated on the {001} facet (c), and two nanocrystals that are truncated on the {112} facet (d and e). Oxygen atoms are shown in red (dark) and titanium atoms are shown in white (light). The arrows in (c-e) represent the relative magnitudes and directions of the nanocrystal dipole moments.
the {001} facets is truncated, so one side is shorter than the Wulff length. This particle consists of 2958 atoms and, as we will elaborate below, the dipole moment is µ ≈ 35 D. The asymmetric nanoparticles shown in Figure 1d,e each contain a {112} facet. Although the {112} facet is not a terminating plane in the Wulff shape, these facets have been observed experimentally23-27 and they have been proposed to play a significant role in the oriented attachment of colloidal anatase.3 The nanocrystals in Figure 1d,e consist of 3141 and 3528 atoms and they possess dipole moments of 250 and 75 D, respectively. We expect the long-range interaction between two TiO2 nanoparticles that are separated by more than a few angstroms to be dominated by the electrostatic (or Madelung) potential. In quantum-mechanical embedded-cluster calculations, for example, the long-range electrostatic potential is often described classically, by a set of point charges placed on the ions.28-33 Here, we use the Matsui-Akaogi force field34 to describe this interaction. In this potential, the interaction energy U between atoms i and j separated by a distance of rij has the form
( )
U(rij) ) Aij exp -
rij Cij qiqj - 6 + Fij rij rij
(1)
The Matsui-Akaogi force field assigns partial charges q of +2.196 and -1.098 to titanium and oxygen, respectively Nano Lett., Vol. 9, No. 12, 2009
Table 1. Parameters for the Buckingham potential of the Matsui-Akaogi Force Field34 ion-ion Ti-Ti Ti-O O-O
A, kcal/mol
F, Å
C, kcal Å6/mol
717654 391053 271719
0.154 0.194 0.234
120.997 290.392 696.941
(qO ) -qTi/2). These effective charges were obtained by Traylor et al. by fitting to the experimentally observed phonon dispersion of rutile.35 In addition to the electrostatic term, the first two terms in eq 1 constitute a Buckingham potential. The parameters for the Buckingham potential, Aij, Fij, and Cij, were adjusted to reproduce the reported elastic constants of rutile, along with the observed crystal structures of rutile, anatase, and brookite.34 The values of these parameters are listed in Table 1. The Matsui-Akaogi potential was shown to be the best out of nine promising force fields compared in a study by Collins and Smith,36 where the properties of interest were lattice energy, polymorphic structures, elastic and dielectric constants, and relative surface energies of titania. The suitability of this potential has been demonstrated in a wide variety of applications,10,37-42 including that of Dubrovinskaia et al., who used it to predict a new high-pressure phase of TiO2 that was confirmed experimentally.42 The Matsui-Akaogi force field has also been shown to perform similarly to a more complex and computationally demanding variable charge model.43-45 4199
Figure 2. Snapshots of the aggregation of the large symmetric nanocrystals. These figures are taken (a) at the beginning of the simulation; (b) after 100 ps; (c) after 160 ps; and (d) after 1.0 ns. Oxygen atoms are shown in red (dark) and titanium atoms are shown in white (light).
To simulate nanoparticle motion, we employed the DLPOLY package,46 version 2.18. We used the Verlet leapfrog algorithm with a time step of 0.5 fs to integrate Newton’s equations of motion. A nanocrystal was constructed and equilibrated for a time period ranging between 250 ps and 2 ns in the canonical ensemble at a temperature of 573 K. For the symmetric nanoparticles, the equilibration was repeated three times using different initial velocity distributions and the results were independent of the initial condition. All nanocrystals essentially retained their initial shapes throughout the equilibration runs. During the equilibration, we calculated the dipole moment µ of each nanoparticle, which is given by µ ) ∑iqijri, where qi and jri are the charge and position vector of ion i, respectively. Because of their symmetric shapes, the dipole moments of the Wulff-shaped nanocrystals in Figure 1a,b are zero. The three asymmetric nanoparticles in Figure 1c-e possess permanent dipole moments, which converged to values of 35, 250, and 75 D, respectively, after a 2 ns equilibration period. Subsequent to equilibration, we replicated the nanocrystals and placed them in different initial configurations corresponding to various center-of-mass separations and orientations relative to one another. We studied 50 different initial configurations for the symmetric nanoparticles and 40 for the asymmetric nanoparticles. Initially, the crystals were always sufficiently far apart that they could rotate freely without touching. We simulated the two-particle systems in the microcanonical (NVE) ensemble. This can be regarded as a realistic environment for vacuum or low-pressure experiments, where heat transfer is slow. The total simulation times in the two-particle runs ranged between 1.0 and 5.0 ns, which is too short to study the entire sintering process, but is long enough to observe the approach, coalescence, and initial restructuring of the nanocrystals after coalescence. The aggregation mechanism shown for the large, symmetric nanoparticles in Figure 2 is representative of that observed in 47 out of 50 total trajectories for the symmetric nanoparticles, that is, in 8 out of 10 for the large and in 39 4200
out of 40 for the small nanoparticles, and in 34 out of 40 total trajectories for the asymmetric nanoparticles. A movie illustrating the final stages of aggregation is available in the Supporting Information. Figure 2a shows the initial configuration, where the center-of-mass separation is ∼8.5 nm. After 100 ps (Figure 2b), the particles have already adopted the relative orientation that they assume upon aggregation. The nanoparticles adopt their final, relative orientation when the shortest distance between two atoms in different particles is ∼7-8 Å for the large nanoparticles and ∼5-6 Å for the small ones. Aggregation begins when an {001} surface of one particle contacts the edge between two {101} surfaces of another particle (Figure 2c). More specifically, one of the oxygen atoms on the edge between two {101} facets initially contacts one of the under-coordinated Ti atoms on the {001} surface of another particle. Subsequently, another O-Ti pair comes into contact and the aggregation occurs in a zipperlike fashion to form a “hinge” that joins the two nanoparticles at/near their edges. Then, one of the particles pivots about the hinge, so that its {101} facet contacts the {001} facet of the other particle (Figure 2d), and a long-time restructuring process ensues. During this long time, the nanoparticles essentially retain their initial shapes, while atoms that reside at the interface between them appear to be slowly rearranging themselves. Thus, aggregation occurs via a “hinge” mechanism, which serves to join under-coordinated oxygen and titanium atoms on the particle facets and edges. It is worth mentioning that in 3 out of 50 simulations for symmetric nanoparticles, aggregation occurred when the {101} surface of one particle contacted the edge between two {101} surfaces of the other particle. After this, one of the particles rotates to bring two {101} surfaces of the particles into contact. To resolve the origins of the observed directional preference for aggregation, we first note that the interaction energy between the two nanoparticles is dominated by the electrostatic term in eq 1. The total electrostatic interaction is always at least 10 times larger than the total Buckingham potential Nano Lett., Vol. 9, No. 12, 2009
Figure 3. The magnitude of the interparticle force on each atom when two small, symmetric nanoparticles are in the initial stages of aggregation. Forces are shown relative to the minimum observed force, which takes the value of one.
in our simulations. Although it is tempting to equate the dispersion-like term in the Buckingham potential with a van der Waals force, the values of the C parameters in Table 1 were fit to bulk properties of TiO2 and are larger than the value of C6 that would be inferred from the Hamaker constant.47 Thus, attractive dispersion forces are negligible compared to electrostatic forces. In Figure 3, we show the relative magnitude of the interparticle force (intraparticle forces are not included in Figure 3) on each atom when the two small, symmetric nanoparticles are in the initial stages of aggregation. Here, we see that the largest forces occur for atoms that direct aggregation, those on the edge between two {101} surfaces of the particle on the left in Figure 3 and an {001} surface of the particle on the right. These large forces occur on twocoordinated O atoms along the edges between two {101} surfaces and for four- and five-coordinated Ti atoms on the {001} surface. In the bulk, Ti atoms are six- and O atoms are threecoordinated, while on the facets, Ti atoms may be four-, five-, or six- coordinated and both two- and three-coordinated O atoms can be found. However, four-coordinated titanium atoms only exist on two edges of the {001} facet, and the two-coordinated oxygen atoms on the edges between two {101} facets have the highest density in the symmetric nanoparticles. The high O density makes aggregation favorable along this edge. Thus, electrostatic interactions between under-coordinated Ti and O atoms drive the aggregation. It is worth mentioning that we monitored the dipole moment as a function of time as the particles approached and aggregated. For all the nanoparticles studied, this quantity remained essentially constant, fluctuating about its permanent value. Interestingly, the asymmetric nanoparticles exhibit similar aggregation mechanisms to the symmetric ones, despite the fact that these particles possess permanent dipole moments. The asymmetric nanocrystals in Figure 1c have a high density Nano Lett., Vol. 9, No. 12, 2009
of two-coordinated oxygen atoms on two of the edges between the {001} and {101} surfaces. While this high density occurs in the symmetric nanoparticles, one of the {001} facets is larger in the asymmetric nanoparticles, so that the length of this edge is comparable to the length of the long edges between two {101} surfaces, which are also rich in two-coordinated O. As for the symmetric nanoparticles, two of the edges between the {001} and {101} surfaces contain four-coordinated Ti. Because of restructuring of the edges of the large {001} facet (which is evident in the curvature of the edges in Figure 1c), the number of fourcoordinated Ti atoms is the same on the edges of the large and small {001} facets and they have a nearly equal propensity for aggregation. Therefore, in addition to exhibiting the mechanism seen for the symmetric nanoparticles in five trajectories, the nanoparticles in Figure 1c also aggregate when two edges between {001} and {101} facets contact in two out of eight trajectories. A movie illustrating the final stages of aggregation via this mechanism is included in the Supporting Information. We did not observe aggregation along the direction of the dipole for these nanoparticles. The asymmetric nanoparticles in Figure 1d,e contain {112} facets, which were observed to be the preferred facets for nanoparticle aggregation in experimental studies of the growth of colloidal anatase.3 Additionally, both these nanoparticles have dipole moments and the nanoparticle in Figure 1d has a large dipole moment (µ ) 250 D) along the {112} facet. Despite this, these nanoparticles aggregated along the dipole moment in only one out of 32 trajectories. In 19 out of 32 trajectories, the aggregation mechanism for these nanoparticles was identical to that for the symmetric ones. Following the general principle observed for the particles in Figure 1a-c, an additional aggregation pathway can be attributed to the high density of two-coordinated oxygen atoms on the edge between the {112} and {101} surfaces in 8 out of 32 runs. In the rest of the trajectories (4 out of 32 total), the aggregation initialized between two {101} facets, or one {101} and one {112} facet. To clarify the additional aggregation mechanism, we created maps of the electrostatic potential surrounding the nanocrystal in Figure 1e. We set up a grid of points in a plane parallel to each nanoparticle facet and obtained the electrostatic potential by summing the electrostatic interactions between a positive charge and the nanocrystal ions at each grid point. In Figure 4a, we see that the electrostatic potential is the most negative around the intersection of the {112} and {101} surfaces and at the edges between two {101} facets, which can be linked to the high density of twocoordinated O atoms in these areas (cf., Figure 4b). As we saw for the symmetric nanoparticles, the four-coordinated titanium atoms are located on the {001} facets, where the electrostatic potential is the most positive. Thus, we observe that anatase nanocrystals aggregate with certain preferred orientations in a vacuum environment. Our simulations indicate that electrostatic forces between undercoordinated atoms on the edges between nanocrystal facets drive this phenomenon. Dipole-dipole interactions have been inferred to be the driving force for directed aggregation in 4201
Figure 4. (a) Projected map of the electrostatic potential around the asymmetric nanoparticle in Figure 1e obtained in planes parallel to and 6 Å from the facets. (b) Coordination number of atoms on the facets and edges of the particle in (a).
some previous studies.1,2,12-14 However, we find that aggregation rarely occurs along the direction of the dipole, even when the permanent dipole moment is as large as 250 D. Although the dipole-dipole interaction is the leading term in the multipole expansion of the electrostatic potential for neutral particles,48 such as those studied here, higher order multipole moments (e.g., quadrupole, octupole, etc.) can also contribute to the electrostatic potential. Our results (cf., Figure 4) imply that these high-order multipoles dominate the electrostatic potential at short separations and that they are created by regions of positive and negative charge associated with under-coordinated surface atoms. Comparing our results to those in experimental studies of colloidal (hydrothermal) anatase,3 we find that our simulated aggregates resemble some of the experimental images obtained with high-resolution transmission electron microscopy. However, in the experimental study, aggregation occurred primarily on {112} facets3 and such events are in the minority here. Moreover, our “final” aggregates are polycrystalline and not the single crystals seen experimentally.3 Given the substantial differences between experiment (colloidal environment, laboratory times) and simulation (vacuum environment, nanosecond times), this discrepancy is perhaps not surprising. Our results indicate that to fully address experimental observations, it is necessary to take into account the aqueous environment surrounding the nanoparticles, which incorporates phenomena such as hydration forces, the dissociation of water, and hydroxylation of the nanoparticle surfaces. It is also possible that restructuring of our initial aggregates into a single crystal could occur over times that are too long for MD, but that are still short for experiments. This possibility could be investigated with accelerated MD simulations.49,50 Nevertheless, the observed mechanism for preferential alignment may be a driving force for oriented attachment and the growth of anisotropic structures during crystallization. Oriented attachment has recently been observed in the vapor-phase growth of ZnO4 and our conclusions may be directly relevant to such systems.
Department of Energy through Basic Energy Sciences Grant DE-FG0207ER46414. Supporting Information Available: Two movies are included to illustrate the “hinge” mechanism for the large, symmetric nanoparticles shown in Figure 1a and for the asymmetric nanoparticles shown in Figure 1c. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)
Acknowledgment. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research and the 4202
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