NANO LETTERS
A Molecular Dynamics Study of the Structural Dependence of Boron Oxide Nanoparticles on Shape
2005 Vol. 5, No. 2 363-368
Susan K. Fullerton and Janna K. Maranas* Department of Chemical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 Received August 18, 2004; Revised Manuscript Received January 3, 2005
ABSTRACT Molecular dynamics simulation is employed to study the effect of varying nanoparticle shape on the structure of boron oxide nanoparticles. Two nanoshapes are investigated and compared: a sphere of diameter 16 Å and a cube of dimension 16 × 16 × 16 Å. A many-body polarization model is employed within the simulation, accounting for dipole moments induced by local electric fields. The resulting network is described by a short-range structure consisting of planar BO3 units, while the intermediate-range structure is described by six-membered planar boroxol rings. Both the fraction of boroxol rings and their locations differ between the two nanoshapes. All planar boroxol rings within the spherical simulation are located on the interior, while planar rings within the cubic simulation aggregate to the cube walls. In addition, structural differences appear between the two shapes at longer ranges, including the formation of “layers” aligned parallel to the walls of the cube, reminiscent of both the low-density crystalline phase and the high-density amorphous form of boron oxide.
Nanoparticles are being considered for application in a variety of areas including but not limited to drug delivery,1,2 biological labeling,3 environmental remediation,4 and as conductivity enhancing additives for solid-state polymer electrolytes.5-7 With increasing emphasis on developing and manipulating nanoscale materials, precise control over the size, chemical identity and shape of nanoparticles must be achieved to further develop innovative nanoscale materials with novel applications. It is well known that size and chemical identity are responsible for many important physical and chemical properties of nanoparticles, including optical and electrical,8 magnetic,9 superconducting,10 luminescence,11 and catalytic activity,12 to cite a few examples. However, the dependence of properties on nanoparticle shape is less understood. One author suggests that shape is as important as size when considering nanocrystals.13 For example, recent findings focusing on semiconductor nanocrystals show that electronic structure and optical properties depend on the ratio of their length and diameter, varying from nanodots to short and long nanorods.14 In addition, observations of properties such as plasmon resonance and electronic states changing as a function of shape have been observed for silver15,16 and CdSe, respectively.17 One factor that limits studies on nanoparticle shape is synthesis. While controlling the size of spherical nanoparticles of various chemical identities has been achieved, controlling and therefore studying nanoparticle shape has presented a challenge. Nonetheless, some nanoparticle shapes * Corresponding author. E-mail:
[email protected]. 10.1021/nl048660f CCC: $30.25 Published on Web 01/25/2005
© 2005 American Chemical Society
of various chemical identities have been synthesized, albeit in mostly small concentrations.18-25 Because shape-controlled synthesis has been met with minimal success, an alternative technique is considered that does not require synthesis: molecular dynamics simulation (MD). A valuable attribute for MD is that nanoshapes can be precisely defined, and structural changes can be monitored with atomic precision, since the exact location and identity of each atom can be observed as a function of time. While all of the experimental studies aforementioned examine the shape of crystalline nanoparticles, this study is unique because results are presented for amorphous nanoparticles. Crystalline nanoparticles can form spontaneously via chemical reactions,4,12,15,20-23,25 whereas amorphous nanoparticles have been created by molding.26 The amorphous nanoparticles in this study are also created by virtually molding the surface atoms into predetermined shapes via confinement forces. Amorphous materials under this type of confinement are particularly interesting, since it has been shown that when the dynamics are modified at the surface, the change persists for a certain distance into the bulk.27 In the present study, the amorphous material under investigation is the network glass former, boron oxide. The goal of this study is to examine how both the surface and overall structure of boron oxide change with nanoshape. Boron oxide (B2O3) is a strong network glass former, according to the classification of Angell.28 The short-range structure of boron oxide is the planar BO3 triangle. Within these triangles, each oxygen is bonded to a boron atom with
Figure 1. Planar boroxol ring: boron [shaded atoms], oxygen [open atoms].
an O-B-O bond angle of 120°. Three BO3 triangles can combine, resulting in the formation of a planar boroxol ring, illustrated in Figure 1. The intermediate range order observed by the formation of boroxol rings differentiates boron oxide from other strong network glass formers. For example, the short range order of the strong glass former, silica, is described by SiO4 tetrahedra; however, intermediate range order is described by random connections of tetrahedra. Boroxol ring fractions of 0.50 to 0.85 have been observed by various experimental techniques, and also appear within molecular simulations that account for polarizability.29-31 In one study, a spherical nanoparticle of boron oxide was simulated, resulting in ring fractions consistent with experimental results.31 It is the existence of the boroxol ring that makes boron oxide an interesting material for the topic of this investigation. First, the number and location of boroxol rings can be observed as a function of nanoshape. Second, boroxol rings are planar, suggesting the possibility that rings might aggregate to planar surfaces if present. This suggestion has been supported experimentally by measurements of boron oxide surface tension, which is one of the lowest for any inorganic glass melt.32 The author argued that planar rings align with the surface, reducing the number of dangling bonds, and thereby reducing surface tension.32 If boroxol rings align with the surface, then it is reasonable to theorize that the nanoparticle interaction with other materials is also a function of shape. For example, it has been shown through MD simulation that the structure and dynamics of PEO near a TiO2 surface differs from that of the bulk, persisting for length scales up to 15 Å.33 In an attempt to isolate a surface effect, the atoms on the TiO2 surface were artificially relocated in the z-direction such that all atoms lie in the same plane (an atomistically flat surface), as opposed to their energetically favored positions. In this variation of the simulation, the structure and dynamics of PEO were equal to that of the bulk. These observations support the concept that atomic surface structure influences the properties of neighboring materials. Therefore, this study aims to examine surface structure as a function of shape, specifically focused on the location of boroxol rings. Considering the experimental and simulation results presented above, the two shapes chosen for investigation include a sphere of 16 Å diameter and a cube of dimension 16 × 16 × 16 Å. The cubic nanoparticle will provide planar surfaces for comparison to a spherical nanoparticle, defined by nonplanar surfaces. In addition, the two shapes share the same three-dimensional confinement, in contrast to a thin 364
film confined in one dimension for example. Furthermore, the dimensions were chosen so that the two shapes could be easily compared in the same manner that nanoparticles of different shapes are compared in the literature.15 We chose the smallest dimension by which the cube can be described, the edge length, as the relevant dimension to compare to the diameter of the sphere. At these sizes, the sphere could fit exactly into the cube, with excess volume occurring only at the corners. Since the goal of this study is to examine the structure of boron oxide as a function of shape, particularly in the presence of flat surfaces, this design maximizes the flat surface area, while retaining a common dimension of 16 Å between both shapes. These shapes are simulated with a polarization model that includes a many-body, nonadditive interaction describing the dipole moments induced by local electric fields. It has been shown in previous publications that while computationally demanding, a polarization contribution must be accounted for in order to obtain results consistent with experiment: specifically the existence of boroxol rings.31,34 The potential energy function is a summation of three distinct contributions dependent on atomic location.31 The function describes boron and oxygen as ions; however, they will be referred to as atoms throughout the remainder of the discussion. The first contribution accounts for pair potentials between atoms; however, pair potentials alone cannot produce polyatomic molecules such as B2O3. Therefore, a many-body potential is added as a second contribution, allowing for the formation of nonlinear and branched molecules, including boroxol rings. This contribution describes the induced dipoles that arise on all polarizable atoms. The solution of an N × N matrix of equations is required every time-step, where N is the number of polarizable atoms in the simulation. This contribution accounts for 99% of the total computation time and makes the boron oxide code 75 times slower than a typical system with more atoms than boron oxide, Coulombic interactions and Ewald summation, but without polarization. Therefore, short simulation times are an unavoidable consequence of the current goal: to investigate structure, specifically focused on the presence and location of boroxol rings, as a function of shape. Mathematical details of the first two potential contributions are described in a previous publication.31 The third contribution enforces particle shape by replacing periodic boundary conditions with conservative confinement forces. These forces serve two purposes: first they provide a virtual “reaction vessel” for the system, and second they confine the system to a specified geometry. The “reaction vessel” is made possible because the forces can be designed to selectively pass or block designated atomic species, permitting the study of reactions with automatic byproduct removal. The geometry of the system depends on the mathematical form of the third potential, and in the case of the sphere it is described by N
ΦIII ) III
(ri/Li)p ∑ i)1
III,Li,p > 0
(1)
where i is the atom number, N is total number of atoms, ri Nano Lett., Vol. 5, No. 2, 2005
is the distance of the atom from the origin, and Li is the radius of the containing wall. The ratio, ri/Li, is a measure of the atom’s location relative to the containment wall. The value of p determines how strongly potential III couples the location of the atom to the wall location. A new potential III was developed for the current study to model nanoparticles that are rectangular in shape rather than spherical. The size of the rectangle chosen by the user is controlled by the constants Lx, Ly, and Lz, representing half the length of the rectangle in each dimension. Since these constants are independent, the nanoparticle may be varied into a rectangle with any aspect ratio desired by the user. For example, the nanoparticle can approach the shape of a thin rod by setting Lx and Ly small and Lz large, or a thin film by setting Lx and Ly large and Lz small. To confine the atoms to the desired shape, each atom must be prohibited from moving beyond the nearest wall. If the distance to the nearest wall is small the potential is large, and if the distance is large the potential is small. Equation 2 gives the potential contribution N
ΦIIIi )
[(
1
∑ i)1 c
1
(δ+x)
1
+
(δ-x)
) ( p
+
1
(δ+y)
+
1
)
p
+ (δ-y) 1 1 + (δ+z) (δ-z)
(
)] p
(2)
where δ+x, δ-x, δ+y, δ-y, δ+z, and δ-z, are the distances to the six walls. Notice that the contributions in each direction (x, y, z) are simply additive. The constant, p, controls how strongly the potential couples the location of an atom to the wall in the same manner as potential III for the sphere. Values for p are chosen for the sphere and the cube so that the potential III contribution is nearly equal for both shapes, as a function of radial distance from the center of each nanoparticle. The spherical nanoparticle was previously simulated and the results are reported in ref 31. Since the polarization model supports reactions, the sample was prepared in the same way that boron oxide is prepared experimentally, namely by the dehydration of boric acid. The density of the spherical nanoparticle was held at the experimental value of 1.84 g/cm3 by setting the “reaction vessel” size equal to 16 Å diameter with 204 atoms. At this density, the system was held at a temperature of 1830 K until network formation was complete. The temperature must be held well above the glass transition temperature of boron oxide so that equilibration can be achieved within the computational time limits. The system was equilibrated for 100 ps, since this was the time required for the mean-squared displacement to reach the diffusive limit, and the self-intermediate scattering function to decay to zero.34 After equilibration, the system was simulated for a production run of 80 ps. It is noted that once network formation was complete, the network did not break up and re-form new network structures at any time during the simulation. The final network contained four planar boroxol rings, located on the interior of the sphere. The cubic nanoparticle was simulated using the potential given in eq 2. In this case, the density was held at the same Nano Lett., Vol. 5, No. 2, 2005
experimental value as the sphere, 1.84 g/cm3; however, the number of atoms were increased to 320 in order to obtain a cube of the desired dimensions (16 × 16 × 16 Å). B2O3 molecules were initially arranged in an ordered array of dimension 24 × 24 × 24 Å, spaced by 2 Å along the two directions parallel to the plane of the molecule with 5 Å spacing perpendicular to this plane. The velocities of all atoms were set to zero initially, and a time step of 0.2 fs was used. Upon starting the simulation, the temperature was set equal to 4800 K. This high initial temperature was chosen to expedite equilibration. During this time, the wall parameters were reduced by equal amounts in each direction until the size of the cube equaled 16 × 16 × 16 Å. The sample was quenched from 4800 to 1830 K in 4 ps at this density - the same quench rate as the sphere. The final temperature of 1830 K was chosen based on results from the spherical simulation, as ring formation was more highly favored at this temperature.31 Since the goal of this study is to investigate structure and since it was previously observed that once the network formed in the sphere it did not break up and re-form, the simulation was stopped and analyzed once network formation was complete. In addition, while the network structure on the surface and on the interior formed simultaneously in both shapes, the rate of network formation in the cube proceeded much faster than the sphere. The fraction of boroxol rings within the systems, calculated as the number of boron atoms located within boroxol rings divided by the total number of boron atoms in the system, equals 0.15 and 0.07 for the sphere and cube, respectively. Experimental results for bulk boron oxide show a decrease of boroxol ring fraction with increasing temperature and a leveling off of the ring fraction at approximately 0.20 for temperatures greater than 1300 K.35 The difference in ring fractions between the two nanoparticles indicate that ring fraction is a function of nanoshape. Even more interesting than the boroxol ring fraction is their location within the nanoparticles. The cubic nanoparticle contains three planar six-membered boroxol rings and one eight-membered ring, while the sphere contains four boroxol rings. All the rings within the sphere are located on the interior of the nanoparticle.31 Considering this information along with the fact that the sphere can fit inside the cube, one would predict that the favored cubic ring location would be the interior of the cube as well. The results indicate otherwise, showing that each ring is positioned on the surface of different cubic faces. The six-membered rings on each of the three surfaces are illustrated in Figure 2. It should be noted that the polarization model does not enforce bonds between atoms; rather, they are drawn for illustrative purposes when two atoms are positioned within a prescribed distance. This distance is set to describe equilibrium bond lengths with allowance for temporary vibrational stretching. It is obvious from these results that rings preferentially aggregate to planar surfaces as opposed to interior networks. It must be made clear that the third contribution to the potential does not attract atoms toward the surface, thereby favoring the possibility of ring formation; rather, atoms pay an energetic penalty for approaching the surface. Therefore, 365
Figure 2. (a-c) Three different sides of the cubic nanoparticle. (d) The spherical nanoparticle. Atoms within six-membered planar boroxol rings are highlighted in black. All rings found within the cubic nanoparticle are located on the surface.
Figure 3. (a,b) Layered structure appearing in two dimensions of the cube where layered atoms alternate between two different shades of gray. (c) The remaining dimension of the cube where no layers are present. In each case, surface atoms on four sides of the cube have been removed for clarity, while surface atoms on the remaining two sides are highlighted in black. Bonds have been omitted to improve clarity.
the formation of rings on planar surfaces is not an artifact of the model. This qualitative observation makes clear that the surface structure of boron oxide is a function of nanoshape, and specifically the presence of planar surfaces. An additional unique structural feature occurs within the cube: the atoms on the interior of the network align parallel to the cubic surface into four “layers” of atoms. The layers are illustrated by a snapshot of the cube in Figures 3a and 3b, where surface atoms on two sides of the cube are highlighted in black and the atoms contributing to the layers alternate between two shades of gray. Surface atoms on the remaining four sides are omitted for clarity. For comparison, Figure 3c illustrates the remaining dimension where no layered structures appear. To quantify this observation, the position density is calculated for both the cube and sphere. The position density describes regions of high and low densities of atoms as a function of distance throughout the nanoparticle. In the case of the sphere, the calculation begins at the origin and records the number of atoms as a function of distance to the edge of the sphere. The cubic calculation begins at one side of the cube and records the number of atoms as a function of distance to the opposite side. The calculation was repeated in the x-, y-, and z-dimensions for the cube. Results for the sphere, and one dimension within the cube (corresponding to Figure 3a) are illustrated in Figure 4. Notice that the spherical data from 0 to 8 Å is simply reflected from 0 to -8 Å for comparative purposes with the cube. It is obvious from Figure 4 that there exists a high density of atoms located on the surface of both the cube and the spherical nanoparticles; however, the feature is sharper for the cube than the sphere. This indicates that more atoms 366
Figure 4. Position density graph for all atoms within the sphere and the cube, normalized by the total number of atoms in each nanoshape. Positions are recorded and plotted as a function of distance from the origin to the radius of the sphere and as a function of distance through the cube in one dimension. No pattern is observed in the other two dimensions for the cube.
are aligned on the surface of the cube than the surface of the sphere, with potential to influence the dynamics of a surrounding material in a similar manner as the TiO2/PEO system previously discussed.33 Even though a large number of atoms are positioned on the surface of both shapes, unlike the sphere, four peaks occur between (5 Å within the cube, indicating the formation of layers. While a high density of atoms is located at (4 Å for both shapes, the cubic structure is uniquely described by an additional high-density area of atoms at (1 Å. These peaks correspond with the layers illustrated in Figures 3a and b. Next we attempt to relate the structure observed in the cubic nanoparticle to amorphous and crystalline forms of boron oxide. The predominating amorph is low-density (1.84 Nano Lett., Vol. 5, No. 2, 2005
g/cm3) amorphous boron oxide, which will be referred to as vitreous boron oxide. Upon pressurization, vitreous boron oxide is transformed to a high-density amorph (2.065 g/cm3). Within this amorphous structure, planar BO3 triangles align into layers, and boroxol rings break apart.36 While both the cubic nanoparticle and the high-density amorphous form share the same layered structure containing BO3 triangles, two important differences remain: the presence of boroxol rings in the nanoparticle and the difference in densities. Therefore it cannot be concluded that changing structure via altering nanoshape is equivalent to changing structure via pressurization. However, before we can attribute the anomalous structural differences solely to the influence of nanoshape, the possibility of pressure-induced structural changes must be addressed. It should be noted that the third contribution to the potential is significant at the wall and quickly vanishes to zero at the origin. Therefore, the average force exerted by potential three at the wall is divided by the surface area of each nanoparticle to calculate an effective pressure. In order for pressure to cause the structural differences, the effective pressure of the cube must be larger than the sphere, since the cubic structure is more consistent with the highdensity amorph. In addition, the pressures must differ by 4 orders of magnitude, consistent with the experimental pressure difference between the low- and high-density amorphs.36 The results indicate otherwise, since the effective pressure on the cube is smaller than the sphere and the difference between the pressures is only 20%. Therefore, the structural differences observed within the cube cannot be attributed to pressurization. We also compare our results to the crystalline phases of boron oxide, since the formation of layers is a characteristic more often observed in crystals than amorphous solids. Two phases of crystalline boron oxide are known to exist: highdensity and low-density. The structure of the high-density crystal is described by interconnected BO4 tetrahedra units with a bulk density of 3.11 g/cm3.37 In contrast, our nanoparticles do not contain tetrahedrally coordinated boron atoms, and furthermore, the density is significantly lower. Instead, the structure of the cubic nanoparticle is more reminiscent of the low-density crystalline phase, which is composed of triangular BO3 units that tend to form layers; however, unlike the cubic nanoparticle, the low-density phase does not contain boroxol rings.38,39 The lattice spacing of the layers within the low-density crystalline phase is approximately 3 Å,40 corresponding to the spacing of the layers illustrated by the position density graph in Figure 4. Despite the similarities between the interior structure of the nanoparticle and the low-density crystalline structure, the nanoparticle density remains less than the crystal density of 2.56 g/cm3.38 In addition, there is no evidence of a structural repeat unit within the layers of the nanopartricle that would strongly indicate a crystalline phase. The cubic nanoparticle structure remains unique when compared to the four forms of boron oxide presented above. It shares a common density and intermediate-range structure (boroxol rings) with vitreous boron oxide, while sharing a layered structure with the high-density amorph and the lowNano Lett., Vol. 5, No. 2, 2005
Figure 5. The (a) oxygen-boron, (b) boron-boron, (c) oxygenoxygen pair distributions for the cube and the sphere. The inset highlights a peak that occurs at 1.26 Å for the cube, where O-B bond distances are slightly shorter than the preferred distance of 1.4 Å.
density crystalline phase. Furthermore, boroxol rings are located only on the surface and not within the layered structure. Apparently the presence of flat surfaces under nanoconfinement permits boroxol ring formation while promoting the formation of layers. Since the presence of well-defined layers differentiates the overall structure of the cube from the sphere, we next determine if packing at short distances is also altered by shape. The packing is described by calculating the pair distribution functions for both nanoshapes, as illustrated in Figure 5. The results for B-B and O-O bonds are similar between the two nanoshapes for distances less than 3 Å. One exception occurs at a distance of 1.26 Å within the B-O pair distribution for the cube. The atoms bonded by this small spacing are located on the surface of the nanoparticles, and the participating oxygen atom is coordinated to only one boron atom (dangling bond). Since the oxygen does not have two network constraints to satisfy, the bond length can be shortened toward the equilibrium bond length of 1.175 Å.31 The peak arises specifically for the cube because when all 367
atoms are considered, a higher percentage of dangling bonds exists in the cube than the sphere. However, when considering dangling bonds that only occur at the surface of the nanoparticle, the dangling bond to surface area ratio is smaller for the cube. This observation demonstrates that dangling bond location is a function of nanoshape and is consistent with the previous hypothesis that flat surfaces reduce the number of dangling bonds on the surface, thereby promoting planar ring formation. At distances longer than 4 Å, differences begin to arise between the cube and sphere, as expected by the position density results. Specifically, a large peak occurs at 7 Å within the sphere that is unobserved for the cube. This peak appears because the geometry of the spherical nanoparticle and the geometries of the largest spherical bins coincide; as a result, all the surface atoms on the sphere are counted in the bins with an approximate radius of 7 Å. Since the geometry of the cube is different, this peak representing surface atoms is unobserved. In closing, the results presented in this study indicate that material structure is a function of nanoshape, specifically for the case of vitreous boron oxide. Currently, the only known method to alter amorphous structure is by pressurization; however, this study shows that structural changes also result from changing the shape of nanometer-sized particles. The results indicate that two amorphous structures share the same density, which differentiates them from previous results for polyamorphism in glass-forming materials. When the nanoshape is altered from a sphere to a cube, the structure changes in two ways: boroxol rings preferentially aggregate to planar surfaces, and aligned layers form between the walls of the cube. The formation of planar rings on the surface reduces the number of dangling bonds, thereby lowering the surface energy. This observation, along with experimental results, supports the theory that the surface tension of boron oxide is low because of surface boroxol ring formation. With respect to the interior structure, the formation of layers is only observed in the cubic nanoparticle and is a characteristic reminiscent of the high-density amorphous form and the lowdensity crystalline phase of boron oxide. Since structural changes are induced by altering nanoshape, it is reasonable to expect that additional properties will be a function of nanoshape. Future investigations should focus on calculating and comparing dynamics of these two nanoparticles, along with simulating additional geometries such as thin films, triangles, and nanowires. Finally, since it has been observed previously that surface structure influences the properties of polymers,33 nanoparticles of varying geometries should be surrounded by a polymer to determine how far into the bulk a nanoshape effect can propagate. Acknowledgment. Financial support from the Department of Energy Early Career Principal Investigator Program (DE-FG02-02ER25535) and the NSF Materials Theory Program (DMR-0074714) is gratefully acknowledged.
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NL048660F
Nano Lett., Vol. 5, No. 2, 2005
