Atomistic Description of Two-Dimensional Hexagonal Close-Packed

Article pubs.acs.org/crystal

Atomistic Description of Two-Dimensional Hexagonal Close-Packed Critical Nuclei Exhibiting Quantization of the Activation Energy Barrier Peter J. Skrdla* 640 Maple Street, Westfield, New Jersey 07090, United States S Supporting Information *

ABSTRACT: A simple methodology is employed to demonstrate the manifestation of energy quantization in classical, steady-state nucleation of small, geometrically precise clusters. With the use of recent dispersive kinetic theory outcomes, fresh insights into nonsteady-state nucleation-and-growth processes are also put forth. The impetus for this work stems from the fact that classical nucleation theory (CNT), which is often a poor predictor of nucleation rates, relies on macroscopic, continuous material properties to describe the physicochemical characteristics of the critical nucleus that, in turn, determines the activation energy barrier for nucleation; those quantities are not physically relevant on the atomic/nanometer scale. While it is shown that the low apparent interfacial tension of the smallest critical clusters likely gives rise to a very diffuse interface, their crystalline core provides a natural high-energy (metastable) state useful in determining the activation energy. For simplicity, it is assumed that the critical nuclei are two-dimensional (2D) hexagonal close packed (hcp) structures without defects and that they can be adequately represented by an ensemble of simple harmonic oscillators (SHOs). As a first approximation, different energies are assigned to three distinct classes of monomers/SHOs present in each 2D hcp nucleus, based on their spatial location within the cluster. The energy of formation for each cluster size, at a fixed chemical potential, is then determined by summing together the energy contributions from all of the SHOs contained therein; the (steady-state) activation energy can then be determined from the maximum in the resulting energy profile. Consistency of the approach with CNT outcomes is demonstrated for larger (∼micrometer-sized), spherical clusters by bridging to traditional, macroscopic properties (surface area, volume, and interfacial tension). Furthermore, it is discussed that while the initial critical nucleus formation at the outset of nonsteady-state (dispersive kinetic) nucleation-and-growth conversions can be a rare event, as the cluster grows into thermodynamic stability, it acquires interfacial tension that serves both to keep it from dissociating and to facilitate its continued growth by attracting additional monomers to its surface. At the same time, larger clusters exhibit a stronger monomer attraction that gives rise to acceleration in both the mean cluster growth rate and the rate of the overall phase transformation. Simultaneously, the system supersaturation, which provides the driving force behind (the initiation of) nucleation, decreases over the course of the conversion. Thus, an apparent paradox of nucleation-and-growth rate-limited conversions is that while the formation of the smallest critical clusters is most difficult (because they have the highest activation energy), they are formed first, while the larger nuclei that have lower activation energies of formation are produced later during the conversion and at a lower supersaturation. The explanation lies in the fact that the larger critical clusters that form at later times are predominantly the growth products of the earlier formed, smallest critical nuclei that survived denucleation, hence the terminology “nucleation-and-growth” for this mechanism. modeling efforts1,14,15 often rely heavily on macroscopic material properties (e.g., interfacial tension, particle radius/ area/volume, contact angle, particle density) to describe the size of the critical nucleus and, hence, to determine the activation energy of (steady-state) nucleation, despite the fact that these quantities do not adequately describe matter on the nanometer scale. For instance, it is well-known that the nonuniformity of matter on the smallest scales, which is not

1. INTRODUCTION Background. Nucleation is a widely studied but still poorly understood process. The goal of accurately predicting nucleation (and denucleation) rates has remained largely out of reach for all but the simplest systems, for many decades. The problem is particularly relevant at present since nucleation ratelimited conversions are important in nanoparticle preparations and1−4 in controlling certain solid-state (polymorphic) interconversions of drug compounds,5,6 and they can potentially inform the dynamics of hydrogel formation.7 Despite the known shortcomings of classical nucleation theory (CNT),8−13 current simulations9 and deterministic © 2013 American Chemical Society

Received: January 4, 2013 Revised: February 23, 2013 Published: March 12, 2013 1970

dx.doi.org/10.1021/cg400020v | Cryst. Growth Des. 2013, 13, 1970−1977

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activation energy, or a time-dependent rate coefficient, all of which simplify to the corresponding, more familiar constants of classical kinetics under steady-state conditions and/or in cases where the conversion rate is significantly slower than the relaxation dynamics. Goals. In the first part of this work, the aim is to describe the relative energies of differently sized critical clusters using only physical quantities relevant on the atomic scale. That is because quantum behavior is to be expected on the nanometer scale, and it is only when the cluster approaches micrometer size that one can treat it in the classical manner.21,22 In the second part of the paper, the findings from the first part are reconciled with both CNT outcomes, applicable to larger clusters under steady-state nucleation conditions, and to previous findings obtained using a DKM for nucleation-andgrowth under nonsteady-state conditions. The System. The selected model system consists of twodimensional (2D) hexagonal close-packed (hcp) clusters composed entirely of atoms that are formed by nucleation from an ideal gas. While the clusters can have different sizes, their morphology is strictly hexagonal (closed shell) and no defects are allowed; this is done for both reasons of simplicity and mechanical viability of the resulting embryonic structures. With regard to the second point, at the single-digit nanometer scale that is of the most interest here, it is thought that any imperfections would result in such a large energetic penalty that the cluster would necessarily dissociate (i.e., denucleate). The smallest nucleus considered in this work contains six atoms arranged in a hexagonal structure around one central atom. Not only does such a structure lack true “bulk” monomer behavior within its core (as per CNT), but determination of the effective surface area (or even the interfacial/line tension) is clearly problematic at these dimensions. In considering the nanometer scale, it is clear that surface tension, in the traditional sense, is a stochastic property of a macroscopic material. In their seminal work on nucleation kinetics, ten Wolde and Frenkel employed traditional, macroscopic physical quantities, discussing that “visual inspection revealed that precritical droplets consisting of only 10 - 25 particles are already quite spherical”.23 However, in a more recent work, similarly treating nucleation of an ideal gas in a Lennard−Jones system, Nowakowski and Ruckenstein intentionally circumvented the use of such quantities (specifically, the surface tension and free energy), in their so-called “kinetic theory of nucleation”,24,25 by determining the critical radius directly from the balance of the evaporation and condensation rates at the cluster surface. Contrastingly, from the author’s perspective, even defining a critical radius in that way might not always be feasible because “the clusters found in simulations are not spherical objects”.26 In support of the current system/approach taken, it is known that 2D nucleation is a common mechanism observed during crystal growth that occurs at supersaturations intermediate between the mechanisms of spiral growth and rough growth.21 Additionally, the observation of real-world nanocrystals (5−50 nm in size) exhibiting a “hexagonal, plate-like morphology”27 is also known from the literature. Due to the structural restrictions put on the 2D hcp clusters in this work, it is evident that the monomer count is quantized and that, in turn, suggests the possibility of quantization in the activation energy barrier of critical nucleus formation. Because defects are not allowed (in either steady-state nucleation or nonsteady-state nucleation-and-growth cases), there inherently

apparent at larger scales, is a direct consequence of the highenergy interface containing many/most of the monomers, as compared to the bulk/interior. The present work is put forth in an attempt to reconcile the quantized nature of matter on the atomic scale with the macroscopic quantities commonly used to estimate the energies of critical nuclei. In previous works,1,6,8,16 the author has demonstrated the application of a deterministic, dispersive kinetic model (DKM) approach to characterize conversions that are rate-limited by the nucleation-and-growth mechanism, operable when the nucleation is not at steady state. The DKM coarse-grains the system dynamics that, in turn, give rise to the formation of differently sized critical nuclei as the system supersaturation is relieved. Since the critical nucleus size and the activation energy of nucleation are linked, via CNT, the DKM shows that the activation energy diminishes over the course of the conversion, as the critical nuclei get progressively larger. The DKM also accurately predicts the experimentally verifiable acceleration in the specific conversion rate, as a function of the time, that has been shown elsewhere to result from an increase in the activation entropy as the conversion proceeds.6,9,17,18 Applying classical (Newtonian) mechanics and basic CNT relationships to the recent findings obtained using the DKM for nucleationand-growth, the rate acceleration was correlated with an attractive force that serves to draw monomers to the nucleus interface; it was furthermore shown that the force is stronger for larger clusters.8,16 In summary, the increase in mean critical cluster size, as the system supersaturation is relieved over time, coincides with an increase in the activation entropy of cluster formation (or, correspondingly, a reduction in the activation energy) that accelerates the rate of the phase transformation; this activation entropy increase and the corresponding strengthening of the “nucleation force” are simply different manifestations (kinetic vs thermodynamic) of the same phenomenon: nucleation-and-growth. DKMs have a distinct advantage when it comes to describing this mechanism, which is two-fold. First, the nonsteady-state behavior discussed here is not inferable from CNT (or models based entirely on that theory). Second, even present-day simulations have insufficient computational power to study dispersive kinetic behavior; due to the fact that nucleation is a low-probability event, it is typically not possible to simulate the formation and growth of more than one cluster in a given system, nevermind to observe a distribution of different critical nuclei over the course of the phase transformation. CNT is akin to a classical kinetic model (CKM) in that it assumes that the system relaxation dynamics are much faster (i.e., they have a much lower activation energy) than the nucleation rate, and therefore, they do not affect the conversion kinetics.6 DKMs are more general than their CKM counterparts (which provide the basis for their derivation) because they do not share the same requirement. In other words, while CKMs consider that the specific rate of conversion is determined by a unique activation energy (corresponding to the CNT case of steady-state nucleation), provided the rate-limiting conversion mechanism does not change over the course of the experiment, DKMs6,19,20 utilize a distribution of activation energies8 or, alternatively, a “manifold of transition states”.9 That activation energy distribution, in cases of steady-state nucleation, simply reverts back to a single-valued activation energy (at fixed chemical potential), as per the CKM treatment. Mathematically, the activation energy distribution can be recast in terms of a distribution of critical nuclei, a time-dependent mean 1971



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useful in showing quantization in the activation energy barrier. Recently, it has been discussed that nuclei emerge as solid clusters having an hcp core embedded within a cloud of surface monomers that are highly correlated with their nearest neighbors but that do not form an ordered crystal structure.26 And while it is widely known that hcp is an atomistic packing geometry that maximizes nearest neighbor interactions while minimizing the space between monomers (each with radius a) since critical nuclei represent an activated state (i.e., a highenergy saddle point in the one-dimensional (1D) energy landscape of possible cluster sizes, see later), it might be possible to consider that the nuclei energies are predominantly linked to the hcp core, with the monomer cloud evolving simply as a direct consequence of the growth of the core. Lastly, as per the treatment of Lovette and Doherty,21 in this work it is assumed that the cluster shape is fixed (at least near the maximum in the free energy landscape, at a specific chemical potential). With regard to the nuclei shown in Figure 1, it was already discussed that in order to maintain the hcp arrangement of atoms yet not incur the high energy penalty associated with (interfacial) defect sites (on the nanometer scale), the cluster formation and growth occurs via the addition of concentric, closed monomer shells around the central atom. On the other hand, any defective nuclei that are initially formed, which are highly unstable, are assumed to dissociate (on the timescale of phonon lattice vibration). Similarly, clusters that form that are smaller than the critical size, at a given chemical potential, will also denucleate. From Figure 1, one can see that the number of monomers in the outermost shell of a 2D hcp nucleus, as a function of the shell number, λ, is given by

exists a geometric selection rule for the monomer count, permitting only complete atomic shells to be deposited sequentially around the central atom (that, by itself, is not considered a possible critical nucleus, as it is in CNT, because its formation does not require an activation energy). It is inferable that the same selection rule might also be useful in describing the nuclei energies. Selection rules are clearly not used in CNT. For one, the Becker−Döring-type28 mechanism describes a continuous, monomer-by-monomer nucleus formation and growth mechanism that does not consider cluster geometry (and, of course, it utilizes macroscopic physical quantities). While that mechanism remains widely used to this day, whereby it is often assumed (for simplicity) that the rates of monomer attachment and detachment are fixed and independent of the critical cluster size,29 both the mechanism and the assumption contradict the author’s recent findings for nonsteady-state nucleation-and-growth8 and the atomistic approach described herein for steady-state conditions.

■

RESULTS AND DISCUSSION 2. Cluster Monomer Count. Consistent with literature evidence of the formation of discrete, atomically precise nanoparticles,30−34 one can consider the 2D hcp nuclei structures depicted schematically in Figure 1 in the development of a new, atomistic treatment of nucleation that can be

nshell = 6(λ − 1) + 6 = 6λ ,

λ = 1, 2, etc.

(1)

The total number of monomers, n, is related by the equation n=1+

∑ 6λ ,

λ = 0, 1, 2, etc. (2)

λ

On the basis of the above two equations, the surface monomer count grows as the series 6, 12, 18, 24, 30, 36, etc., while the total monomer count trends as 7, 19, 37, 61, 91, 127, etc. for the corresponding values λ = 1, 2, 3, 4, 5, 6, etc. The number of monomers is therefore quantized, with the selection rule requiring an integer value of λ. That point made, this selection rule is, of course, not necessary to relate the discrete (atomistic) nature of the nuclei shown in Figure 1, but it allows both a convenient and a potentially physically relevant description of the clusters, for the reasons discussed above. Equation 1 can be further specified to differentiate between monomers at “edge” (corner) sites and those that are uniquely incorporated into the “facets”. The appropriate relationships are as follows:

Figure 1. Schematic diagrams of the two smallest 2D hcp nuclei comprised of atoms of radius, a, where λ = 1 and λ = 2 relate structures containing one and two atomic layers around the central atom, respectively. The angle defined by Ψ/2 = 60° relates the six-fold axis of symmetry of the clusters. The relationship between the cluster interfacial/line tension, ζs, and the “monomer attractive tension” at the grain boundary, ζgb, is graphically depicted in the diagrams (see text for details).

nedge = 6

(3)

n face = 6λ − 6

(4)

The number of monomers in the interior/“bulk” of each nucleus is therefore related by the equation nbulk = 1 +

∑ 6λ λ− 1

(5)

3. Energy Quantization, Cluster Energy Landscape, and the Nucleation-and-Growth Mechanism. To obtain 1972



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where kB is the Boltzmann constant, T is the absolute temperature, and Fn of the nucleus can also be defined as follows, using the partition function, Zn, of the n-mer:

the relative energies of the various 2D hcp nuclei depicted in Figure 1, without using macroscopic quantities, the clusters are considered to be composed entirely of simple harmonic oscillators (SHOs) whose energy levels, Ev, are quantized according to the equation ⎛ 1⎞ Eν = ⎜ν + ⎟ℏω = 0, 1, 2, etc. ⎝ 2⎠

Fn = n faceμface + nedgeμedge − nbulk μ bulk = −kBT ln(Zn) (12)

Following from the derivation of ten Wolde and Frenkel,23 Zn for 2D nuclei can be defined as follows:

(6)

where ν is the quantum number, ω is the angular frequency, and ℏ (≡ h/2π) is the reduced Planck constant. Note that eq 6 also holds for coupled harmonic oscillators that are often used to describe the phonon modes of crystals. If all of the oscillators are assumed to be in the ground state (as per nucleation near room temperature) and neglecting intercluster interactions (while under mild conditions the density of clusters can be relatively low due to the fact that nucleation is a low probability event; that is not the case for most nanoparticle syntheses whereby large supersaturations are used to generate huge quantities of small clusters, almost simultaneously, in order to minimize their subsequent growth, hence, achieving quasisteady-state nucleation conditions), ν = 0 and Ev = 0.5ℏω for a free SHO. To incorporate the effects of spatial location within the crystal lattice, as a first approximation, it is assumed that the energy of the edge site SHOs is fixed and independent of the cluster size. Similarly, bulk sites and face sites are considered also to have fixed energies. That is because all bulk monomers have six, equivalent nearest neighbors while face sites have only two such neighbors located at the interface, plus two more in the bulk. Moreover, monomers found at face (but not edge) locations are considered to have twice the energy of those in the bulk because they have half the number of surrounding neighbors to stabilize them. Contrastingly, since an atom at a face site sits flush with its neighbors (Ψ = 180°), while one located at an edge makes a 120° degree angle with its nearest neighbors at the interface, the energy of an edge site is considered to be 180°/120° = 1.5-fold greater than that of a face site. Resultantly, the relative contribution to the net cluster energy of the different monomer (SHO) types can be expressed as a function of the cluster size by utilizing the earlier monomer count equations: ⎛ ℏω ⎞ ⎟ = (1 + ∑ 6λ)⎜ ⎝ 2 ⎠

(7)

⎛ ℏω ⎞ ⎟(2)(1.5) = 9ℏω μedge = (6)⎜ ⎝ 2 ⎠

(8)

μ bulk

λ− 1

⎛ ℏω ⎞ ⎟(2) = 6(λ − 1)(ℏω) μface = (6λ − 6)⎜ ⎝ 2 ⎠

Zn =

∑ e −F / k T n

B

(13)

n

where A is the area occupied by the cluster, m is the monomer mass, and Λ is the thermal de Broglie wavelength, defined as h 2πmkBT

Λ≡

(14)

However, on the nanometer scale, A also exhibits quantization in the present case (e.g., for λ = 1, it is assumed that A ≈ 7πa2). Thus, one can recast eq 13 as follows for the 2D hcp nuclei using the proposed selection rule for λ: Zλ =

2πa 2(1 + ∑λ 6λ)3 Λ2(1 +∑λ 6λ)(1 + ∑λ 6λ)!

∑ e−{[−(1 +∑

λ− 1

×

6λ)( ℏω )] + [6(λ − 1)(ℏω)] + (9ℏω)} 2 /kBT ,

λ

λ

= 1, 2, etc.

(15)

From eq 15 one can write: ⟨Nλ⟩ = e−ΔFλ / kBT

(16)

corresponding to eq 11. If one assumes a closed system containing both n monomers and N particles at any given time, with negligible pressure variation during steady-state nucleation, then the Helmholtz free energy can be replaced by the Gibbs free energy (without the need to consider PV work) as per ⟨Nλ⟩ = Ne−ΔGλ / kBT

(17)

The intensive eq 17 relates the probability of formation of a cluster of a given nucleus size, as determined by λ, relative to the probability of forming N clusters of various sizes. Using eqs 3−5 for the n values and eqs 7−9 for the μ values, the relative Gibbs free energy change (at a fixed net chemical potential) for the formation of the first seven 2D hcp nuclei, in order of their size, is given by

(9)

For a 2D hcp nucleus formed via ideal gas−liquid nucleation, the Helmholtz free energy change, ΔF, is defined by the following equation for a cluster containing n monomers (where n = nface + nedge + nbulk): ΔFn = (n faceμface + nedgeμedge − nbulk μ bulk ) − nμ

An2 Λ2nn!

(10)

The average number, , of clusters of size n in the system is therefore related by ⟨Nn⟩ = e−ΔFn / kBT = e−[(nfaceμface + nedgeμedge − nbulk μ bulk ) − nμ]/ kBT (11) 1973

⎛ ℏω ⎞ 17ℏω ⎟ + 0 + 9ℏω = ΔG(1)|N , T , P , n = −⎜ ⎝ 2 ⎠ 2

(18a)

⎛ ℏω ⎞ 23ℏω ⎟ + 6ℏω + 9ℏω = ΔG(2) = −7⎜ ⎝ 2 ⎠ 2

(18b)

⎛ ℏω ⎞ 23ℏω ⎟ + 12ℏω + 9ℏω = ΔG(3) = −19⎜ ⎝ 2 ⎠ 2

(18c)

⎛ ℏω ⎞ 17ℏω ⎟ + 18ℏω + 9ℏω = ΔG(4) = −37⎜ ⎝ 2 ⎠ 2

(18d)

⎛ ℏω ⎞ 5ℏω ⎟ + 24ℏω + 9ℏω = ΔG(5) = −61⎜ ⎝ 2 ⎠ 2

(18e)


Crystal Growth & Design ⎛ ℏω ⎞ −13ℏω ⎟ + 30ℏω + 9ℏω = ΔG(6) = −91⎜ ⎝ 2 ⎠ 2

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As per a comment made by a second journal reviewer, it is noted here that the activation energy for (steady-state) nucleation, which is useful for determining the rate constant of nucleation, is typically much higher than the activation energy associated with most monomer diffusion constants; indeed, that is what makes the processes considered in this work nucleation rate limited and not diffusion rate limited (i.e., it is the substantial difference in the two activation energies that allows one to separate the mechanisms). Using Figure 2, one can assume that λ = 6 (n = 127) is the smallest thermodynamically stable embryo. On the other hand, if λ = 3 is taken to be the (only) activated state species (critical nucleus), by assigning the 37-mer an activation energy of formation21 of ∼35kBT, one obtains an estimate for the SHO vibrational frequency of ∼1.9 × 1013 Hz at room temperature. While that value is approximately three-fold (∼ 70/23) larger than the dissociation frequency, kBT/h, of the transition state species predicted by the Eyring equation, it is pointed out here that nucleation is not a chemical reaction but rather a type of phase transformation mechanism. Additionally, that difference should be considered in light of the “10 orders of magnitude”10 error frequently attributed to the pre-exponential term (which consists of the product of the rate of attachment of monomers to the nucleus, the Zeldovich factor, and the number density of monomers) of the nucleation rate equation predicted by CNT, keeping in mind the simplicity of the current approach that does not allow one to identify specific lattice (acoustic phonon) modes/surface wave anisotropy that can give rise to lower frequencies that also might be important for nucleation (nucleation-and-growth). The energy profile depicted in Figure 2 is very different from the (similarly time-independent) activation energy distribution determined elsewhere for the nucleation-and-growth mechanism, as related to the author’s DKM,8 in that Figure 2 exhibits only a single activation energy (near the maximum in the plot) because it reflects steady-state nucleation. Applied to nonsteady-state (dispersive kinetic) cases, Figure 2 would reflect only a given “snapshot” in time, whereas the activation energy distribution function previously determined applies to all times during the phase transformation. Thus, under nonsteady-state conditions, one can essentially observe a different version of Figure 2 (with different critical nucleus size/“instantaneous” activation energy) to hold at every point in time (i.e., each different supersaturation level) during the phase transformation, indicating that both the critical nucleus size and the cluster size needed to achieve thermodynamic stability are essentially “moving targets”. This idea is depicted schematically in Figure 3, whereby it is shown that the initial critical clusters that form at the higher supersaturations are smallest in size and, correspondingly, have the higher activation energies. It is the summation/integration over all of the individual energy landscapes at the various chemical potentials (taking into account also the rate at which the specific activation energies are sampled during the conversion that is governed by the relaxation dynamics), shown in Figure 3, that ultimately gives rise to the activation energy distribution predicted for the DKM of interest.6,8,16 As mentioned in the Introduction, under nonsteady-state conditions one can expect to observe a rate acceleration of the overall phase transformation that reflects a decrease in the mean activation energy as a function of the time/extent of conversion. The time-dependent variation in activation energy or, equivalently, the variation in the activation entropy, in the

(18f)

⎛ ℏω ⎞ −37ℏω ⎟ + 36ℏω + 9ℏω = ΔG(7) = −127⎜ ⎝ 2 ⎠ 2 (18g)

Interestingly, the λ = 1 (n = 7) and λ = 4 (n = 61) clusters have the same energy, just like the λ = 2 (n = 19) and λ = 3 (n = 37) 2D hcp nuclei. That (unexpected) observation, at least to some extent, might reflect the simplistic nature of the present treatment (specifically, the assumptions and approximations discussed above). On the other hand, the flatness of the energy profile in the vicinity of the maximum shown in Figure 2 is

Figure 2. Gibbs free energy of formation, plotted as a function of λ, for the 2D hcp nuclei shown in Figure 1 at fixed chemical potential. The inset shows the relative contributions to the free energy (●) due to the Ψ = 120° edge sites (△), the 180° facet sites (▲), and the interior monomers (○); see text for details. Note that if ℏω ≈ kBT, the 1D free-energy profile shown gives rise to similar values on the ordinate as plotted in ref 26.

supported by the findings of Lechner et al.26 who reported that “the size of the crystalline nucleus is not sufficient to describe the transition...”. Furthermore, due to the quantized nature of the cluster energies, it is not possible in this case to define the critical nucleus size (at steady state) simply by evaluating the maximum in the plot of ΔGλ versus λ (shown in Figure 2) because the function is not continuous, as per the critical radius function in CNT.28,35−37 Similarly, the Zeldovich factor, which relates the width of the activation energy barrier in the vicinity of the critical nucleus38,39 in CNT, also warrants redefining to account for the quantization. Regardless, as pointed out by an anonymous journal reviewer, this paucity in the energy landscape might help explain previous observations of clusters of very specific sizes (so-called “magic clusters”). It is highlighted here that while, according to the present treatment, it is not possible to generate a cluster size smaller than λ = 1, CNT permits monomer-sized nuclei that seem to be more consistent with the mechanism of spinodal decomposition than that of nucleation. Regardless, the general shape of the plot in Figure 2 is consistent with CNT predictions for larger, spherical clusters (discussed more in the next section). In addition, that energy profile also mirrors curves obtained for arbitrarily shaped 2D nuclei, as determined elsewhere.21 1974



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the corresponding (macroscopic) value of the bulk material. Due to that effect, it is of interest here to investigate the Tolman length,42,43 δ, which is defined as the difference between the equimolar Gibbs dividing surface (GDS), re, and the surface of tension in the planar limit, rs: δ ≡ lim re , rs→∞(re − rs)

(19)

For spherical clusters (note that for disks, the numerical coefficient becomes unity in the equation below), as the size decreases, the interfacial tension, γ, deviates from the macroscopic/planar value, γ∞, according to the well-known relationship: ⎛ ⎞ 2δ γ = γ∞⎜1 − + ...⎟ re ⎝ ⎠

Figure 3. Schematic depiction of the dynamic evolution of the Gibbs free energy (ΔGr) profiles relevant to the formation of spherical clusters, shown as a function of both conversion time, t, and the cluster radius size, r (e.g., that has a mean value of ∼10 nm for a small drug molecule undergoing phase transformation via a nucleation-andgrowth mechanism on the minute/hour timescale at near-ambient temperature1) under nonsteady-state conditions. While the individual plots are intended to generally mimic the one shown in Figure 2, the small r values and the large r termini of each curve were selected arbitrarily. Note that t and r have a nonlinear relationship, which is presented elsewhere,1 depicted here by the different directionality of the arrows. The critical cluster radius and corresponding activation energy, for each curve, are defined by the coordinates of the maxima in each plot. As nucleation-and-growth proceeds over time, the critical nucleus becomes larger as the system supersaturation (pressure) is relieved due to monomer consumption by the condensed phase. At the same time, the mean activation energy becomes smaller (because the activation entropy increases, depicted by the width of the curves), thus accelerating the conversion rate. The broken line is an arbitrary depiction of the point where ΔGr = 0, below which any clusters that form are thermodynamically stable.

While it is thought that use of the Tolman length is valid down to very small droplet sizes,44 its use, just like the interfacial tension, can present a challenge in describing matter on the atomic scale, not to mention the deviations from spherical geometry that can accompany shrinkage of the cluster down to the dimensions treated earlier in this work. Regardless, it is because of the interfacial tension lowering45 that, for example, de Aguiar et al.46 found a minimum approximate order of magnitude increase in the interfacial area occupied by surfactant (SDS) molecules at the n-hexadecane droplet/water interface for particles having a radius of only ∼83 nm (and, of course, the cluster sizes considered herein are generally smaller than that). Similarly, the same effect led Horsch et al.38 to implement a surface property correction to CNT in order to account for “the lower surface tensions of small nuclei and their nonsphericity”. While eq 20 is only an approximation, it is possible to infer from it that interfacial tension effectively only “begins” at the point where δ < re/2 (consequently, γ → 0 as δ → re/2). On the basis of this observation, one can infer that the initial clustering mechanism, giving rise to the formation of the first/ smallest critical nuclei (at the highest supersaturations), is generally a stochastic, thermal density fluctuation-induced event. Back to the case of the 2D hcp nuclei, instead of surface tension one must consider the linear tension. Simplifications of the Herring equation,39,47,48 which describes the equilibrium between interfacial forces at a triple line, are known to be useful in defining analytic expressions for the tensions involved at a grain boundary. One such simplification is that of Mullins,49 which is given by the equation below, for 2D cases and it is applied here to the nuclei shown in Figure 1:

system supersaturation or in the critical nucleus size, is reflected in the distribution of activation energies that spans all laboratory times. Two things are worthy of consideration here: first, it is possible also to simulate the evolution of various subpopulations (critical nucleus sizes) in the global distribution, at different points in time.40 Second, the changes in those macroscopic parameters (both kinetic and thermodynamic) have been discussed to reflect an enhancement in the nucleation force, on the microscopic scale, that was presented previously for the case of large clusters,8 and it is clarified in the next section for the tiniest 2D hcp nuclei. 4. Transcending Nucleus Size Scales During Nucleation-and-Growth. In the limit of large clusters, the 2D hcp nuclei can be expected to acquire a disklike morphology as the edge and face sites approach the same energy limit, ultimately producing a smooth surface with a specific interfacial tension that is due to continuous structural rearrangements at the interface driven by minimization of the interfacial area. As λ → ∞, the cluster radius, r, can be approximated as ⎛ 1⎞ r ≈ ⎜λ + ⎟a ≈ λ a ⎝ 2⎠

(20)

ςgb ςs

⎛Ψ⎞ = 2 cos⎜ ⎟ = 1 ⎝2⎠

(21)

In eq 21, ζgb is the linear tension of the grain boundary extending away from the nucleus and, using symmetry, ζs is the linear tension along the interface. Since the angle defined by Ψ = 120° for 2D hcp nuclei of a given size, ζgb = ζs. It is thought that the attractive potential resulting from ζgb serves to draw nearby free monomers to the six, highest-energy edge sites along the perimeter of the (initial) postcritical nucleus. As the (critical) cluster grows in size during system relaxation (provided nonsteady-state conditions), both the interfacial tension and cluster size increase, giving rise to a progressively stronger attractive force that was discussed

(19)

whereby one can clearly observe the discretization in r. In three-dimensional (3D), the disk shape gives way to a spherical cluster, for which the Gibbs−Thomson (Kelvin) equation can be used to relate the activation energy dependence on r.1 However, as the spherical particle is shrunk down to the nanometer scale, the interfacial tension is lowered relative to 1975



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recently in terms of the latter.8 The simultaneous evolution of the interfacial tension was presented here, for the first time, in the above discussion. For small 2D hcp nuclei, the attractive force is directed toward the edge sites that resemble thermal grain boundaries in a bulk material. However, as the cluster grows and its interface becomes smoother and its morphology more 3D, its associated potential not only strengthens, but eventually it extends both uniformly and radially around the cluster (for sufficiently large, spherical nuclei).8 The resultant parabolic potential is akin to that of a SHO or, on the macroscopic scale, to that of an elastic material (whereby Young’s modulus simply replaces the force constant in the former description). Given the dynamic nature of the nucleation force,8 the observation of others that every hcp core has associated with it a cloud of monomers that grows in larger, postcritical clusters26 appears to be supported. Moreover, in common nanocrystal (NC) preparations, it is understood that the larger particles tend to aggregate more readily than smaller ones (which aids in their size-selective precipitation)50 due to their stronger van der Waals interactions.51 For more discussion about the nucleation force, please see the Supporting Information. A rigorous physicochemical link between the continuous evolution of the activation energy, as a function of the conversion time6,8 during (dispersive kinetic) nonsteady-state nucleation-and-growth, and the quantized nature of the activation energy barrier presented here for steady-state nucleation remains elusive. That is because the former effect, depicted schematically in Figure 3, is due mainly to the activation entropy (which is time dependent), while the latter (classical, time independent) case reflects predominantly the activation enthalpy, as per Figure 2.

localized, out-of-equilibrium concentration gradients that, in turn, serve to stimulate the onset of nucleation in neighboring regions and, soon thereafter, throughout the entire system. The other contributing factor to the rate acceleration observed in nucleation-and-growth conversions (which gives rise to their sigmoidal transients) is the strengthening nucleation force that results from the formation of increasingly larger critical clusters, with stronger interfacial tensions, at longer conversion times.8 From an orthogonal perspective, the rate acceleration can also be attributed to a lowering of the activation energy barrier of critical nucleus formation as the system supersaturation is relieved. Once all of the supersaturation is relieved, a different mechanism (e.g., crystal growth or aggregation, etc.) can take over as rate-limiting, in determining the longer term phase evolution. A commonly used approximation, based in CNT, is the socalled “capillarity approximation”. It correctly predicts that higher system supersaturations give rise to the formation of smaller critical nuclei. However, it simultaneously predicts a lower activation energy barrier for the formation of those smaller clusters. It warrants clarification here that the latter result is valid only as applied to the maximum in the cluster energy profile. By applying dynamical (DKM type) considerations or, alternatively, in considering the entire potential energy landscape of the differently sized clusters (not just the maximum), it is clear from the Gibbs−Thomson (Kelvin) equation that smaller particles having a larger degree of curvature have a correspondingly higher activation energy of formation than larger clusters.1,8,41 Resultantly, an interesting paradox of nonsteady-state nucleation is that while it is harder to form a small critical cluster than a large one, because the Gibbs free energy of activation is higher, it takes longer, on average, to generate a larger critical nucleus than a smaller one because nucleation-and-growth necessitates the formation of small clusters before big ones.1,16

5. CONCLUSIONS The smallest critical nuclei, formed at the highest supersaturations (monomer pressures), cannot be rigorously described using traditional, macroscopic physical quantities such as surface area, volume, interfacial tension, contact angle, etc. At the same time, they make prime candidates for investigating the manifestations of quantization in condensed matter on the nanometer scale, as well as for providing new insights into (early stage) nucleation kinetics. Using the simple case of 2D hcp nuclei, quantization of the activation energy barrier was demonstrated in atomically precise clusters formed and grown via the addition of consecutive monomer shells, whereby each monomer within a given cluster was treated as an independent SHO. Ignoring the quantization, which selects only for specific cluster sizes, the energy landscape determined in this work was found to closely resemble the behavior predicted by CNT for the steady-state nucleation of larger clusters. The approach taken herein is thought to be easily extendable to 3D geometries and to various monomer packing arrangements (e.g., via the use of the monomer half-crystal position39 in place of the positional assignments of the 2D SHOs presented in this work to determine the monomer count and assign corresponding monomer energies). While nucleation is often considered a low-probability, fluctuation-induced event, the formation of the initial critical nucleus that is of the smallest size (at the highest supersaturation, early in the conversion, under nonsteady-state conditions) results in a cluster that has minimal interfacial tension to keep it from dissociating. However, irrespective of its ultimate fate (growth or denucleation), its formation induces

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ASSOCIATED CONTENT

S Supporting Information *

Additional discussion regarding the nature of the nucleation force and its role in dispersive kinetic modeling is provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The author declares no competing financial interest.

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REFERENCES

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Atomistic Description of Two-Dimensional Hexagonal Close-Packed

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