DOI: 10.1021/cg101088d
Surface Area Limited Model for Predicting Anisotropic Coarsening of Faceted Nanoparticles
2011, Vol. 11 158–165
Alireza Seyed-Razavi,†,‡ Ian K. Snook,*,† and Amanda S. Barnard‡ †
Applied Physics, School of Applied Sciences, RMIT University, GPO Box 2476, Melbourne, Victoria 3001, Australia, and ‡CSIRO Materials Science & Engineering, Private Bag 33, Clayton South, Victoria 3169, Australia Received August 18, 2010; Revised Manuscript Received November 12, 2010
ABSTRACT: The engineering and control of systems of monodispersed polyhedral nanoparticles are important for the development of devices based on size and shape-dependent properties. This includes a detailed understanding of how these designer nanoparticles are formed, and how they evolve and change over time. Although an experimental strategy or atom based theory may be used to observe different stages of this process, it is highly desirable to model the entire process, based on a limited set of physical or chemical parameters. Here we derive a kinetic model of nanoparticle shape evolution as a function of time, using specific concepts relating to surface diffusion, rate of step growth, and particularly surface site availability, referred to as the surface area limited (SAL) theory of growth and nanomorphology. This model is then applied to examples, including the axial growth of an arbitrary nanorod and the anisotropic growth of a nanogold truncated octahedron, restricted to either the Æ111æ or Æ100æ direction, each driven by the coarsening mechanism. Introduction Nanoparticles and other low-dimensional nanostructures are currently the subject of intense scientific interest, and nanotechnology has already made important contributions to nanocatalysis,1-7 electronics,8 photonics,9 information storage,10,11 imaging,12,13 sensing,14-18 and various biomedical applications.19-24 However, the next step in the advancement of this technology is to achieve more sensitive control over the structure and properties of individual nanomaterials, both during formation and when in use. To achieve this, it is highly desirable to understand the evolution of nanoparticles (both formation and aging) and ultimately use this knowledge to engineer precise characteristics for specific applications. However, the evolution of nanoparticles is a complicated issue, and exploring this issue experimentally is very challenging, particularly since nanoparticles exhibit a high degree of morphological anisotropy. When synthesizing nanoparticles from solution (or the gas phase), following nucleation, the subsequent formation of an individual nanostructure occurs by two primary mechanisms: growth involving aggregation25 and growth involving coarsening.26 In the latter case, there are many factors that influence the outcome, such as size,27 initial “seed” morphology, temperature,28,29 chemical environment,30 and the addition of surfactants/capping agents.31 These factors must be explicitly included in any formal description of coarsening. As this is a multidimensional problem, a theoretical model, capable of describing coarsening, ripening, aggregation, and coalescence, and with sufficient accuracy so as to be comparable to experiments, would be invaluable in this regard.32 It is also essential that such a model be capable of describing faceted nanoparticles, as well as nanorods, nanowires, and other more complex nanostructures. To date there have been a number of models derived to represent this type of growth. Classical theories, utilizing *Phone: þ61-3-9925-5290. Fax: þ61-3-9925-2143. E-mail: ian.snook@ rmit.edu.au. pubs.acs.org/crystal
Published on Web 12/08/2010
methods of continuum thermodynamics and kinetics, have been used to describe the evolution of discrete structures occupying the nano- to microscales (and beyond). With the advancement of computational power and efficiency, these classical models were either expanded upon or surpassed by higher accuracy atomistic and molecular methods, which rely on molecular dynamics (MD),33,34 or probabilistic considerations, such as the random walk models of kinetic Monte Carlo (KMC) methods.35 However, although these types of studies are very instructive, they all fall victim to a number of inherent limitations. First, each simulation can be limited with respect to the number and types of shape (motifs) that are accessible during the time of the simulation, depending on the initial starting configurations and sizes, and whether the kinetic barriers to morphological transition are breached. It is entirely possible that alternative shapes and structures may exist (and be more stable) but, given the initial configuration, would only emerge if the simulations were to be run for a prohibitively long time. The testing of all possible starting configurations would eliminate this problem, making the time frames required more reasonable, but will necessitate a large number of unsuccessful attempts.32 To overcome these limitations, we turn here to less computationally expensive theoretical models which, although approximate, still incorporate the essential physical processes involved in nanoparticle growth. In this paper, we present a new, general, surface area limited model for nanoparticle evolution via coarsening, which is both consistent with established classical theories but facilitates the modeling of faceted nanoparticles and anisotropic nanostructures using a limited set of input parameters (which may be calculated using first principles simulations or measured experimentally). This model effectively provides a multiscale kinetic theory of nanoparticle shape evolution and can be used for any material under a range of conditions and with any starting configuration (nucleus or “seed”). To illustrate how this may be achieved, we apply the model to two simple cases including the axial growth of an arbitrary nanorod and r 2010 American Chemical Society
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the anisotropic growth of a nanogold truncated octahedron restricted to either the Æ111æ or Æ100æ direction. Theoretical Background The theoretical bases for our surface area limited model are the classical theories derived by Lifshitz, Slyozov and Wagner36,37 and the Burton, Cabrera and Frank model,38 and they assume that the driving forces of growth are dominated by (1) the diffusion of monomers from the surrounding bulk to the nanoparticle and (2) the surface diffusion occurring on the facets of a nanoparticle of a given shape. In the first instance, the well-known Ripening model devised by LifshitzSlyozov,36 and later expanded upon by Wagner (LSW),37 is, to this day, considered to be one of the fundamental theories used for the analysis of the rate of growth of spherical particles (or “grains”), described via the radial evolution as a function of time. The Lifshitz-Slyozov theory incorporates the concept of a cluster at equilibrium with its surrounding vapor, but the vapor is itself not in equilibrium, which is referred to as a diffusion limited theory (eq 1a). Soon after, Wagner introduced the opposing case, in which the vapor is in equilibrium but the cluster/vapor interface is held in a nonequilibrium state; this phenomena has been termed the reaction limited theory (eq 1b). dR DS 2γVm 2 C0 1 1 ¼ ð1aÞ dt ÆRæ R R kB T dR DS 2KγVm 2 C0 ÆRæ 1 ¼ dt ÆR2 æ ÆRæ R kB T
ð1bÞ
where R is the radius, ÆRæ is the mean radial distribution, DS is the surface diffusion coefficient, γ is a parameter which contains the interfacial surface tension and atomic volume of the solute, K is the step growth rate, Vm is the molecular volume of the species, C0 is the concentration of the vapor or solution, kB is Boltzmann’s constant, T is the absolute temperature, and t is time. Both incorporate the Gibbs-Thomson relation39 and utilize Fick’s law of diffusion, and an in depth analysis can be obtained from refs 40-42. In the second instance, the greatest conceptual insight given by the Burton, Cabrera, and Frank (BCF) theory is that the growth at the surface of a crystal, or any nanostructure, is heavily influenced by the existence of steps and kinks, which provide preferable sites for the attachment of adatoms (or growth monomers). This step-flow theory results in a rate equation of the growth of steps (K) on a crystal surface which evolves either as a basic step island or as a spiral: Vm 2DCeq j¥ ð0Þ ð2Þ K ¼ xs a0 where Vm is the molecular volume, a0 is defined as the lattice spacing, D is the bulk surface diffusion coefficient, Ceq is the equilibrium concentration of monomers on the surface, j¥(0) is a dimensionless variable which is dependent on the differences between the surface and bulk concentrations, and xs is the mean distance diffused by an adsorbed atom. A detailed explanation of the BCF model is provided in ref 43. These models have proven very successful in explaining the coarsening of spherical nanostructures, but the remaining challenge is to effectively combine these concepts to describe the evolution of finite nanostructures decorated by planar facets or different crystallographic orientations, which coarsen
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at different rates due to surface diffusion and different adsorption characteristics. This may be achieved by considering each “single facet” as an independent surface of an arbitrary size, within a supersaturated reservoir of monomer concentration (C0). We will show that, in such a system, the important influences include the gradient of the chemical potential between the nanoparticle and the surrounding reservoir, as described by the Gibbs-Thomson relation. Surface Area Limited (SAL) Model Derivation We begin by deriving a rate equation which describes coarsening as a change in the length of the nanoparticle, L, perpendicular to a given surface facet, i, with respect to time, t. This rate equation will be of the following form: dLi Ki Xd ¼ Neff , i dt
ð3Þ
where Neff,i is the effective number of sites available for monomers to adsorb and possibly diffuse, X is the number of monolayers present of thickness d (where d is defined as the thickness of a single monolayer), and Ki is the rate of formation of a single monolayer on facet i. On a given facet, with side lengths a and b, there exist Nt,i = na nb number of possible adsorption sites, where na and nb are the number of sites along sides a and b, respectively. In real experiments, not all of these sites are accessible to the growth monomer, and surfactants often interact strongly with particular sites on the facet (chemisorption), effectively blocking them from the monomers and restricting coarsening. In general, if NP sites are passivated by a surfactant in this way, then Neff,i = Nt,i - NP,i where Neff , i 0e e1 ð4Þ Nt, i When coarsening is restricted to one facet due to complete passivation of competing facets, then either the surface area of that facet and Neff,i are constant or decrease, at the expense of new facets being introduced. However, when coarsening is unrestricted, the relative area of each facet (and Neff,i on each facet) changes after each monolayer deposition, depending on the degree of anisotropy. This term appears in the denominator, since a larger number of sites will take longer to fill, so the rate of single monolayer deposition is slower. However, as X increases (more monolayers are deposited), the value of dLi/dt increases, even when the number of available sites is constant and the rates remain unchanged. This is illustrated in Figure 1, showing simple examples of restricted, anisotropic coarsening on different facets of a hypothetical tetragonal nanoparticle (shown via the transparent extensions). In this example the number of sites (depicted by transparent spheres) on each coarsening facet is constant, but we can see that the sizes on neighboring facets change, such that: Nt, x ¼ na nb Nt, y ¼ nb nc Nt, z ¼ na nc
ð5Þ
Returning our attention to eq 3, the rate of growth of a monolayer, Ki, is given by eq 2 and may be written in terms of the surface flux JS = 2DSCeqj¥(0),43 such that Ki ¼
Vm JS, i a0 xs, i
ð6Þ
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Figure 1. Schematic representation of the increase in facet lengths na, nb, and nc when coarsening is restricted to different facets, and how this influences the number of effective sites available for adsorption. Growth is illustrated by transparent volumes in different directions, including (a) the initial particle before coarsening, (b) coarsening parallel to c, which provides additional sites in directions a and b, (c) coarsening parallel to a, which provides additional sites in directions b and c, and, finally, (d) coarsening parallel to b, which provides additional sites in directions a and c.
From Fick’s law of diffusion we know that JS, i ¼ - DS, i rμi
ð7Þ
The diffusion coefficient for monomers on the facet may be approximated by ð8Þ DS, i ¼ va0 2 e - Ediff , i =kB T where kB is Boltzmann’s constant and T is the absolute temperature at which this process occurs, ν is the sum of the vibrational frequencies of the monomer on the surface (and the rotational frequencies if required), and Ediff is the activation energies required for diffusion. This is related to the hopping rate κs,i = Ds,i/a0. The mean distance diffused, xS, can also be expressed as a function of the activation energy of adsorption (Ead) and the activation energy required for desorption (Edes), such that xS ¼ a0 eðEad, i - Edes, i Þ=2kB T ð9Þ At this point, we highlight that this model assumes that diffusion occurs across the surface, and it is therefore applicable to gas phase systems. It is not generally applicable to solutions, as monolayer deposition is not assured, and deposited growth monomers may not equilibrate their surface positions with the same (or a comparable) diffusion rate. Moreover, since a single surface diffusion process is included at this time, this version of the model is restricted to isotropic surfaces and is ideally suited to metals.
Equation 7 also introduces the chemical potential gradient rμi, since the Gibbs-Thomson relation states that the chemical potential of a particle increases with decreasing particle size. This is because the equilibrium solute concentration near a small particle is higher than that near a large particle, resulting in molecular transfer between either a group of nanoparticles or a nanoparticle and a reservoir of monomers. The driving force for the movement of atoms across the solid/bulk interface is the differences in chemical potentials between the surface and the surrounding vapor or solution. Therefore, by combining eqs 2, 7, and 9, we obtain KS, i rμi ð10Þ Ki ¼ - Vm ðE - E Þ=2k e ad, i des, i B T where the chemical potential gradient rμi has the form Ceq ðtÞ ð11Þ rμi ¼ kB T ln C0 ðtÞ which is a function of the concentrations of the reservoir C0(t) and the concentration of monomers participating in the coarsening monolayer, Ceq(t). Defining this concentration in terms of the number of effective and available adsorption sites, we find that Ceq ðtÞ ¼ Neff , i - Ni ðtÞ
ð12Þ
where N(t) is equal to the number of available sites remaining for successive adsorption at a time t. If the reservoir is infinite,
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C0(t) = constant. If the reservoir is finite, then the change in the two concentrations with respect to time would be equivalent and, therefore, constant: dCeq ðtÞ dC0 ðtÞ ¼ ¼ const ð13Þ dt dt It naturally follows that Ceq(t) = C0(0) - C0(t), but alternative definitions of C0(t) may be used to introduce replenishment of the reservoir, either via Ostwald ripening or via some external mechanism. Finally, by combining eqs 3, 8, 10, and 11, we obtain a general equation for the rate of coarsening of facet i, with respect to time: 0 1 E C ðtÞ diff , i eq B C kB T ln Bv exp C dLi B C0 ðtÞ C Xd kB T " # B C ¼ - Vm B CNeff , i dt ðEad, i - Edes, i Þ @ A exp 2kB T ð14Þ which is related to the bulk flux of monomers through the reservoir to the surface, given by JB, i ¼ Jeff ¼
dLi Neff , i dt Vm
ð15Þ
This model assumes that the area of each facet is finite and that it will remain so during the time frame under consideration. Thus, the coarsening is limited by the finite surface area of the facets, which results in the surface area limited (SAL) model for the coarsening of individual facets of any polyhedral nanostructure (particle, rod, belt, or wire), that may be parametrized for any material, monomer, or surfactant. It requires the calculation or measurement of Ead, Edes, and Ediff and the appropriate selection (or variations) of Ceq(t), Neff, C0(t), and T. The remaining parameters are naturally defined by the system in question. The model will output the size at a given time t, with its characteristic shape, which has evolved up until that time. If a nanostructure is enclosed entirely by the same type of facet (as for example an octahedron enclosed by {111} facets), then the model describes isotropic growth when the distribution of surfactants is also isotropic (or surfactants are absent). If the nanostructure is enclosed by a combination of different facets or the distribution of the surfactant is anisotropic (or both), then the model describes anisotropic growth and faceted (or even irregular) nanostructures. At this point, it is important to point out that the definition of the unique facet i does not necessarily correspond to a crystallographic orientation {hkl}. In homoelemental systems or nonpolar materials, this may well be the case, but surface polarity can render facets with the same indices inequivalent, so i (hkl) is more general. Application to Model Systems To demonstrate the capabilities of this simple model, let us consider the epitaxial growth of a gold nanorod, oriented in the Æ001æ direction, and enclosed by passivated {110} facets perpendicular to the principal axis.44 In this case, due to the passivation of the {110} facets, growth is restricted to the (001) facets, which have a limited surface area, and coarsening is epitaxial. In this direction, we have calculated the values of Ead = Edes = 1.28 eV and Ediff = 0.78 eV, using density functional theory within the generalized gradient approximation
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via the PBE exchange correlations functional45 and projected augmented wave (PAW) pseusopotentials46 as implemented by VASP code.47 This method has been successfully used to model the surface properties of gold in the past.48-50 This leaves the choice of Ceq(t), Neff, C0(t), and T to be tested. In these examples, full partition functions have not been generated for the diffusion process, and surface phonons are omitted at this stage. In Figure 2 we show results calculated using eq 14 for T = 300 K and n = 6 (Neff = 36), beginning with a cuboid seed, which corresponds to a gold seed ∼2 nm in average diameter, with an aspect ratio R = 1, and with 1/3 of the surface area accessible to the monomer (shown in red in Figure 2) and 2/3 of the surface area blocked by a surfactant (shown in green and blue in Figure 2). The facets that do not contribute to the growth process are those that have been passivated. One can consider the work of Niidome et al. and the CTAB surfactant process, which results in the growth of anisotropic gold nanorods.51 If we assume an open system, where the concentration of the reservoir is a constant, we may explore the role of Ceq(t) by setting it to some constant values (Ceq). As we can see from Figure 2a, a small concentration of monomer at the surface gives rise to rapid coarsening, characterized by the slope of the aspect ratio as a function of time, since the gradient of the chemical potential is high. In the space of 1 h, the nanorod approaches an aspect ratio of R = 4. As the value of Ceq increases, the coarsening slows, and when Ceq is unity, R < 3 after 1 h. When Ceq = 1000, R < 2 after 1 h. The individual steps in these graphs are related to the formation of individual monolayers, and we can see that as Ceq increases, more time is spent forming each layer, as the flux to the surface is reduced. As a consequence of the coarsening in the axial direction, and the observed change in R, there is also a change in the relative fraction of surface area associated with the available facets and the facets passivated/blocked by surfactant. This is shown in Figure 2b for the same values of Ceq(t), Neff, C0(t), and T, where we can see how changing Ceq(t) controls the relative fraction of different available/unavailable facets after a given period of coarsening. If we assume a closed system, where the concentration of the reservoir diminishes as monomer is adsorbed onto coarsening nanoparticles, the results change very little until the reservoir becomes significantly depleted. In reality, however, the value of Ceq(t) is dynamic and will depend on the amount of monomer adsorbed after a given period. This may be self-moderated, where the value of Ceq(t) accumulates over time (Ceq(t) = Ceq(t - 1) þ Neff dL(t - 1)/dt), moderated by the accumulation on a collection of nanoparticles (for example Ceq(t) = Ceq(0)/t), or driven (Ceq(t) = Ceq(0)/exp{-krt}, where kr is a first order rate constant). Results for each test case are provided (for the same gold nanorod) in Figure 3, showing the bulk flux of monomers to the surface (Figure 3a and eq 15), the effect these Ceq(t) functions have on R (Figure 3b), and the associated effect on the relative fraction of available/unavailable facets. We can see that although Jeff varies considerably, R is only different (lower) when coarsening is self-moderated, and the relative fraction of available/unavailable facets is largely unaffected. In these examples, coarsening is diffusion limited, which occurs when the monomer reacts instantly with the surface, and Ceq(t) , C0(t). In such examples, coarsening in the only unpassivated direction is expected, but we can see how the rates change as the chemical potential changes (as the monomers
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Figure 2. Examples of (a) the aspect ratio and (c) the fractional areas for three possible static values of Ceq, and schematic representations of the axial coarsening of a nanorod from a cuboidal seed, restricted to the available (red) facet and blocked on the remaining (blue and green facets).
Figure 3. Examples of (a) the effective flux, (b) the aspect ratio, and (c) the fractional areas for three possible dynamic functions of Ceq(t).
are removed from the reservoir and deposited on the surfaces). The chemical potential gradient is negative, and the coarsening of the nanoparticle is controlled by the bulk diffusion of the monomer to the surface. dL/dt decreases as more monolayers are deposited, and is therefore size-dependent. When Ceq(t) = C0(t), the chemical potential gradient is zero, and dL/dt = 0. It is also possible to model coarsening under reaction limited conditions, when Ceq(t) . C0(t), and the
chemical potential gradient is positive. In the future, we intend to implement comparable simulations where a stochastic function is introduced to moderate the adsorption/desorption of surfactants, so that alternative facets are only partially blocked. This introduces a fractional value of dLi/dt on the partially blocked version of facet i but requires that equations be solved self-consistently, which cannot be done at this time.
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Figure 4. Shape evolution of a gold nanoparticle enclosed by {111} and {100} facets: (a) when {111} facets are blocked by surfactants, and coarsening is restricted to the Æ111æ direction at 300 K, (b) the same shape evolution as a function of temperature in the range 200-420 K, and (c) schematic representation of the visual appearance.
It is also important to point out that the lower limit of the model is strictly imposed by the initial nucleus size (seed) that is input by the user. It is not possible to dissolve below that size, but if the chosen seed is smaller than the critical nuclei size, then growth does not occur, or more correctly, the addition of the initial monolayer takes an infinite amount of time. Under reaction limited circumstances, while the coarsened material will be allowed to dissolve and return the monomers to the reservoir, the initial seed is ultimately recovered and hence the entire structure does not dissolve. And finally, we note that the coarsening of the nanoparticle is controlled by the reaction of monomers on the surface, and dL/dt is independent of the amount of monolayer deposited. In this case, amorphous materials are usually formed, which is beyond the capabilities of the present model, since we assume the surfaces and any additional monolayers are crystalline when calculating the materials properties Ead, Edes, and Ediff. Since amorphous materials can form truly spherical or cylindrical nanostructures (whereas crystalline materials form polyhedra), a spherical approximation is appropriate for reaction limited coarsening, and one may refer to the work of Sugimoto52 or more recently by van Embden et al.53 Application to Restricted Coarsening of a Polyhedral Gold Nanoparticle In each of the examples described above, coarsening is onedimensional and restricted to a facet that does not change in size. This is, for the most part, the same as epitaxial growth of a single crystal thin film and is relatively rare in nanoscale systems. One of the beneficial aspects of this model is that it enables the treatment of polyhedral particles where the areas of facets, and even the existence of facets, changes over time. It can model the shape evolution, in addition to the size evolution of the system. To demonstrate this, we turn to the example of a single crystal quasi-zero dimensional gold nanoparticle. The low
energy surfaces of gold are known to be perpendicular to the Æ111æ and Æ100æ orientations, so the shape may occupy the geometric continuum between the dual Platonic solids: the octahedron and the cube. An octahedron is entirely enclosed by {111} facets, and the cube (or regular hexahedron) is entirely enclosed by {100} facets. In between the number of stationary points is prescribed by the number of combinations of lattice planes in the Æ100æ and Æ111æ directions, because the material is crystalline. The choice of seed/nucleus shape is arbitrary, so we will begin with the shape with 50% {100} surface area and 50% {111} surface area. Using our definition of surface area in terms of the number of available surface sites, and due to the differential in-plane packing densities, this equates to 43% of sites distributed evenly on sites on the six {100} facets and 57% of sites distributed evenly on the eight {111} facets. From this seed, it is possible to select surfactants that will selectively passivate either the {100} or {111} facet, thereby restricting coarsening to the opposing direction. This sends the nanoparticles along a different evolutionary trajectory, shown in Figures 4 and 5, respectively. In Figure 4 we can see that when coarsening is restricted to the Æ111æ direction, the {111} facets diminish in size, and the area of the {100} facets increases. Initially, a single monolayer makes a dramatic difference to the relative fraction of {111}/{100} surface area (Figure 4a). As the coarsening proceeds, the relative difference is less prominent, until the new octahedral corners are formed and the shape is entirely enclosed by {111} facets. We also see that these new corners are unstable, and fluctuations occur as the shape oscillates between 100% {111} (when the corner site is filled) and 90% {111}/2% {100} (when the corner site is available). When the temperature is altered, the time required to reach the morphological conclusion (or form any of the intermediary shapes) changes, as shown in Figure 4b. Coarsening is much more rapid at higher temperatures, and we can see how temperature can be used to tailor nanoparticle growth.
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Figure 5. Shape evolution of a gold nanoparticle enclosed by {111} and {100} facets: (a) when {100} facets are blocked by surfactants and coarsening is restricted to the Æ100æ direction at 300 K, (b) the same shape evolution as a function of temperature in the range 200-420 K, and (c) schematic representation of the visual appearance.
In Figure 5 we can see that when coarsening is restricted to the Æ100æ direction, the area of the {111} facets increases, while the {100} facet decreases. Once again, the change in the relative fraction of the {111}/{100} surface area is most obvious at the early stages (Figure 5b) and the coarsening concludes when the new cubic corners are formed, and the shape is entirely enclosed by {100} facets. We can see by comparing Figures 4a and 5a that the rate of coarsening in the Æ100æ direction is almost identical to that in the Æ111æ direction, as the lower diffusion barrier (calculated to be 0.154 eV by using DFT, as described above) is balanced by the in-plane packing density. In the case for the Æ111æ direction, coarsening is hastened by an increase in temperature (see Figure 5b), but unlike the case of the octahedral corners, no fluctuations are observed. Relationship to Experiment As we can see from the examples above, although the currently version of the model has some restrictions in terms of applicability and is ideally suited to studying isotropic particles such as metals, there are numerous features that are common to experiments. These include the dependence on the surrounding temperature (that may also be extended to include thermal fluctuations and phonons in the nanoparticle), the stable binding of surfactants to the surface (that may be allowed to desorb and resorb via a suitable temperature- and adsorption energy-dependent stochastic function), appropriate time-dependence and realistic growth rates,54 and realistic polyhedral morphologies coupled with atomic scale resolution. Based on the common features, it is possible to discuss how this model relates to some experiments reported in the literature. First, our model can be used to evaluate the relationship between the shape and the size of the initial seed, as both are related to the number of effective growth sites. In a recent report, Jana described how sizes of monodispersed quasispherical gold nanoparticle seeds (between 1.5 nm, 3.5 nm, 5 nm, 8 nm, and 12 nm in diameter) could be used to produce
anisotropic gold nanorods in micellar templates55 and demonstrated that the aspect ratio and the size of the seed are intrinsically linked. Nanorods obtained when the smallest 1.5 nm seeds were used had an aspect ratio between 1 and 5. Larger (5 nm) seeds failed to yield many nanorods, irrespective of the seed concentration used. If the seed size is larger than 5 nm, the nanoparticles grow in bulk solution, as the template mechanism does not work and results mainly in symmetric nanoparticles or fractals, in the case of restricted growth. Our use of a seed as input to the model is highly applicable in this regard. It was pointed out that the synthesis of anisotropic nanoparticles on the 1-100 nm length scale remains challenging, as the tendency toward isotropic shapes is strong, and the relative rates of nucleation and growth become very important. If the nucleation rate is too high, coarsening ceases before symmetry breaking occurs and quasi-spherical particles are produced. If the growth rate is too slow, the nanorods grow in two or three dimensions and again lose their anisotropy, or the micelle template fails entirely.55 An intermediate and optimum nucleation-growth rate is essential for an effective template mechanism, and the model described here can assist in finding what that optimal rate might be (for a given situation) and how it is affected by surfactants. Moreover, although not discussed, the rate of formation of partial monolayers can also be useful in understanding the evolution of rough surface facets or those decorated with specific types of functional kinks and terraces. Finally, as mentioned above, future work is planned to model the unrestricted growth, where coarsening is allowed in all directions simultaneously (modeled self-consistently), to predict the time required to produce a specific shape, in addition to the above-mentioned nonzero probability for adsorption/desorption of the passivating surfactants. This is particularly important, as the formation of different shapes has been attributed to the interplay between the faceting tendency of the stabilizing agent and the characteristic growth kinetics.56-59 For example, cetyltrimethylammonium bromide
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or CTAB molecules (a common surfactant used in the synthesis of gold nanoparticles54) binds more strongly to the {100} than the {111} faces,60 so the treatment of different facets on polyhedral particles (that go beyond the spherical approximation) is imperative.61 Conclusion In this work, a kinetic surface area limited (SAL) model has been developed, to go beyond the spherical approximation and explore the evolution of faceted nanostructures. The model is consistent with the classical theories of Lifshitz, Slyozov and Wagner,36,37 and Burton, Cabrera and Frank,38 is complementary to the work of Sugimoto,52 and is based on a limited set of physical and chemical parameters that may be measured or computed using electronic structure methods. We have demonstrated the use of this simple model, by predicting the time evolution during epitaxial growth of a gold nanorod oriented in the Æ100æ direction and the morphological evolution of a faceted gold nanoparticle enclosed by the {111} and {100} facets. These examples show how restricted growth may proceed, when alternative facets are passivated by surfactants, and work is underway to model unrestricted growth, where a nonzero probability of surfactant desorption is introduced and coarsening is allowed on all facets simultaneously. Future work is also planned to expand the model to describe Ostwald ripening and more experimentally relevant terms to describe the relationship between the evolving nanostructure and the surrounding reservoir.
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