Use of Dispersive Kinetic Models for Nucleation and Denucleation to

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Use of Dispersive Kinetic Models for Nucleation and Denucleation to Predict Steady-State Nanoparticle Size Distributions and the Role of Ostwald Ripening Peter J. Skrdla* 640 Maple Street, Westfield, New Jersey 07090, United States ABSTRACT: Fundamental kinetic understanding of the formation of various particle size distributions (PSDs) and the time evolution of the mean particle size can guide new synthetic approaches, or improvements to existing ones, for obtaining a desired nanoparticle (NP) morphology, size, and monodispersity. Previous modeling efforts have focused largely on classical kinetic descriptions of nucleation, growth, and particle coarsening/Ostwald ripening (OR) mechanisms, employing numerical methods to simulate the temporal evolution of the NP PSDs. In a very different approach, the activation energy distributions corresponding to recently derived dispersive kinetic models for nucleation and denucleation (Skrdla, P. J. J. Phys. Chem. A 2011, 115, 64136425) are utilized in this work to derive analytic functions for the stationary/ steady-state PSDs relevant to each mechanism. Additionally, the same models are used to obtain the time evolution of the mean NP radius. PSDs for these nanometer-scale phase transformation mechanisms have not been predicted previously in the literature using such a direct approach, circumventing the need for stochastic simulation. The predicted PSD shapes, used individually, together, and/or in combination with the known stationary PSD shapes relevant to OR, are used to qualitatively establish the mechanisms giving rise to PSDs reported in the recent literature. Using this approach, the origin of bimodal PSDs and the phenomenon of PSD focusing are explained. Moreover, the time-evolution functions for the mean NP radius predicted by each mechanism are shown to be sufficiently different so as to allow the three mechanisms to be readily distinguished from one another in treating empirical data.

’ INTRODUCTION Advancements in nanotechnology are driven in large part by the development of controlled material structures that have dimensions on the nanometer scale.1 For instance, semiconductor nanocrystals (NCs) or quantum dots (QDs), including certain transition metal (e.g., Cd, Zn, Pd) sulfides/selenides/tellurides2 and metal oxides,3 have been the focus of much recent work due to their usefulness in such applications as light-emitting diodes4 and as biological labels.5 The phenomenon of quantum confinement6 is key to those applications. Similarly, metallic nanoparticles (NPs)7 whose size approaches the Fermi electron wavelength can exhibit interesting properties that make them attractive for use as catalysts,8 in magnetic storage media,9 and for use in various biomedical applications.10 In the pharmaceutical industry, the increasing demand put on achieving biorelevant exposures of poorly water-soluble drugs has stimulated interest in the development of so-called 00 nanoformulations00 ;11 alternatively, the use of NPs (e.g., silica) as drug carriers has also been discussed.12 At present, there is a limited understanding of the relevant mechanisms (and their associated kinetics) that determine the formation and evolution of NP distributions. That is unfortunate because detailed knowledge of those processes can guide synthetic approaches aimed at tuning the desirable physicochemical properties of NPs. The ultimate goal is to control the NP size,13 monodispersity,14 and shape,15 as those can be determining r 2011 American Chemical Society

factors,16 e.g., in the formation of larger structures (using the NPs as 00 building blocks00 17) in the next level of material hierarchy.7 This work attempts to clarify some of the most important mechanisms thought to be involved in current NP preparations— nucleation, denucleation (the complementary process to nucleation), and particle coarsening (Ostwald Ripening, OR)—their kinetics, their limiting particle size distribution (PSD) shapes, and the various functional forms for the time evolution of the mean particle radius that they predict. In terms of modeling nucleation/ denucleation kinetics, current approaches are often complex, and they predict conversion rates with only limited accuracy; that is why the (deterministic) models introduced in this work are founded on dispersive kinetic principles (vide infra). With regard to PSD control, many workers have relied on a very small number of classical models, including LaMer’s model of burst nucleation,1821 to guide their synthetic approaches. In general, NP formation is thought to start with the formation of a critical nucleus, as defined by Classical Nucleation theory (CNT),2226 which assumes steady-state nucleation conditions (i.e., a constant monomer supply/supersaturation level). It is known that, during the process of nucleation, any nuclei formed that are smaller than the critical size will be unstable and thus Received: August 21, 2011 Revised: November 28, 2011 Published: November 29, 2011 214

dx.doi.org/10.1021/jp2080513 | J. Phys. Chem. C 2012, 116, 214–225

The Journal of Physical Chemistry C dissociate, while those that are larger than the critical nucleus size will continue to get larger due to the fact that NP growth is thermodynamically favorable beyond the critical size. The critical nucleus size and its corresponding work of formation are directly related to the Gibbs free energy of activation for nucleation, via CNT equations, and hence so is the specific nucleation rate. While CNT readily predicts many experimentally verifiable behaviors, including faster nucleation rates at lower temperatures (i.e., at higher monomer supersaturations), it is a poor predictor of nucleation kinetics.2729 Some of the limitations of CNT can be traced back to the assumptions inherent in the theory.2932 The use of dispersive kinetic models is thought to address some of those limitations. The field of dispersive kinetics is based on the fundamental concept of an underlying distribution of activation energies,3340 which might account for energy quantization.4143 Using the assumption of a distribution of activation energies (or, equivalently, a time-dependent activation energy42), coupled with a first-order conversion mechanism, the author has been able to precisely describe the sigmoidal conversion vs time (x t) transients associated with many nucleation and denucleation rate-limited solid-state conversions.44 Such transients, which are generally asymmetric (hence they exhibit either 00 accelerating00 or 00 decelerating00 character45), have been recently discussed by the International Confederation for Thermal Analysis and Calorimetry (ICTAC) to result from 00 distributed reactivity00 . Consequently, they require the use of a distribution of rate coefficients/ activation energies to describe the kinetics,45 consistent with the dispersive kinetic treatment presented herein. Modeling/predicting PSDs and the t-evolution of the mean NP size can be daunting tasks because of the host of different mechanisms that can be involved during the course of a given synthetic process. Complicating those efforts is the fact that different mechanisms can apply to different types of NPs (e.g., insulators, semiconductors, or conductors) synthesized using various experimental procedures/conditions/precursors/surfactants (e.g., hot injection,2,3,4648 two-phase reduction,49 solgel processing,50 00 hot organometallic00 routes,7 photochemical reduction,51 etc.). For instance, it has been reported that metallic colloids usually form via a nucleation-and-growth mechanism (involving the formation of nuclei larger than critical size, immediately followed by growth via the direct consumption of monomers from solution),7 while OR is often dominant in the preparation of semiconductor colloids (consequently, it is commonly utilized in that role for tuning the mean particle size52,53). Furthermore, recent works have raised compelling evidence of other mechanisms that might be important to consider in certain cases, e.g., NP aggregation/coalescence5460 (which remain under investigation61). As Robb and Privman noted, 00 it is experimentally challenging to unambiguously quantify the size distribution of nucleated nanoparticles because of their tendency to aggregate, their distribution of nonspherical shapes, the onset of Ostwald ripening, and other factors00 .62 In contrast to the approaches taken in the literature to date, the use of dispersive kinetic models to predict NP PSDs is much simpler. That is because such models directly account for system heterogeneity that classical deterministic models (similarly having closed form, analytical solutions) cannot achieve. Furthermore, the mathematics involved in the PSD prediction are simpler (and, hence, more readily accessible to experimentalists63) than numerical approaches/simulations that provide only a general stochastic treatment of the underlying

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mechanisms, often under the assumption of thermodynamic quasi-equilibrium.46,48,62 The dispersive kinetic models for nucleation and denucleation presented herein44 are simple and easy to use, and they provide a ready interpretation of the key physicochemical aspects of the kinetics via only two semiempirical parameters.30,42,44,64 In a recent work, the activation energy distributions relevant to those models were defined.42 In the present paper, a unique and attractive property of those models is leveraged to define the PSDs corresponding to nucleation and denucleation mechanisms: the ability to directly correlate the underlying activation energy distribution function to a t-invariant/stationary PSD function. Using the PSD shapes predicted by those models, both individually and in combination, the underlying mechanisms giving rise to certain experimentally observed PSD shapes will be qualitatively explained. The key objective of this paper is to provide a simple theoretical basis, utilizing the author’s two dispersive kinetic models and classical OR relationships, for the observation of various PSD shapes (including those that are asymmetric and tail either to the right or to the left, as well as symmetrical Gaussians and bimodal distributions) and different trends in the t-evolution of the mean particle size that are commonly encountered in the literature. That has not been possible before using traditional models, without the use of numerical methods to simulate stochastic behavior.

’ RESULTS AND DISCUSSION 1. Limitations of CNT and the Case for Dispersive Kinetic Models. According to CNT, bulk properties are assumed inside

the nucleus despite the fact that metals and semiconductors of nanometer dimensions exhibit quantum confinement behavior suggesting that quantum mechanical rules should be substituted for the laws of classical physics.65 Furthermore, the interfacial tension is assumed to be independent of nucleus size, despite knowledge to the contrary that has led to the use of correction factors such as the Tolman length66 (which are also ignored here, for simplicity). More importantly, because CNT predicts only one critical radius size (i.e., a unique activation energy) for any given set of (steady-state) experimental conditions, it does not consider that nucleation might be a dispersive kinetic process, i.e., that the nucleation rate can be influenced by the system dynamics and thus give rise to a distribution of activation energies/t-dependent activation energy. In other words, noninstantaneous, out-of-equilibrium relaxation effects can produce a continuous evolution in the activation energy as a function of the nucleation time/extent of conversion. After all, nucleation is now widely understood to be a complex nonequilibrium process whereby any fluctuations of the critical nucleus can lead to either its decay or growth.67 Additionally, it is clear from classical thermodynamics that if steady-state conditions are not maintained in the nucleating system the energetic requirements for critical nucleus formation change over time (e.g., as supersaturation is relieved). For instance, Peters68 recently used a time-dependent 00 superaturation schedule00 in his modeling of isothermal nucleation kinetics, following the earlier work of Kashchiev and Firoozabadi.69 In a similar respect, the author’s dispersive kinetic models42 have built into their functionality consideration of the 00 lag time for reorganization of the nucleus size distribution00 68 via the assumption of a distribution of activation energies.43 215

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The Journal of Physical Chemistry C

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It has been observed experimentally and through computer simulations that synthonic structures (dimers, trimers, etc.), which are thought to lead to critical nucleus formation, evolve temporally during dissolution and crystallization events.7074 Those findings support the use of a t-dependent activation energy in modeling nucleation/denucleation rate-limited conversions, providing direct support for dispersive kinetic models. Moreover, Anwar and Zahn recently acknowledged32 that nucleation likely occurs as a result of a 00 manifold of transition states00 (i.e., a distribution of activation energies75). In that regard, coarse-grained modeling approaches,76 which include the dispersive kinetic equations presented in this work, are attractive in that they simultaneously treat both the kinetics and dynamics in the simplest possible manner. 2. Background and Goals. The earlier modeling work by Sugimoto21 provided important insight into the formation of a single colloidal particle in a fixed environment. Despite its applicability to the micrometer scale, Peng et al.77 were able to use that theory to describe the PSD narrowing (focusing) and broadening (defocusing) of their IIVI and IIIV semiconductor NPs, linking the former process to a steady-state supersaturation level produced by a balance between free monomer production (via the precursors used in the synthesis) and their consumption via NP formation/growth. More recently, Talapin et al.46 extended that modeling effort by describing the time evolution of NP PSDs utilizing CNT, coupled NP growth/ dissolution and OR mechanisms. Two years ago, van Embden et al.48 published a NP PSD modeling work describing much the same mechanisms but using a different numerical approach, claiming to have used more experimentally relevant quantities in their simulations than those found in other theoretical papers published over the past decade. Their paper indicated that the final crystal sizes are generally not much larger than the primary nuclei, suggesting a 00 strong overlap between nucleation, growth, and coarsening00 during early reaction times—noting that their simulations also generally showed a separation in time between the NP nucleation/growth and OR mechanisms, with an intermediate 00 quasi-equilibrium00 state thought to generate the monomer concentration needed for eventual coarsening. In this work, the limiting PSD shapes and the t-evolution of the mean NP radius size, predicted directly from the author’s dispersive kinetic models for nucleation and denucleation, are introduced and used in conjunction with the classical results predicted for OR to yield a simple explanation for the origin of common PSD shapes/NP size evolution kinetics that can be found in real-world experimental data. The author’s dispersive kinetic models for nucleation and denucleation are thought to be just as useful in describing the formation78 of NP PSDs as they are in certain, more traditional solid-state phase transformations that are also rate-limited by critical nuclei (including crystallizations, crystal dissolutions, and thermal decompositions, as well as polymorphic interconversions44). While dispersive kinetics can be observed in all phases of matter and across vastly different time scales,44,79 empirical evidence of activation energy distributions relevant to NP formation kinetics might be found in certain works.55 Furthermore, while the recent observation80 of smaller (on average) iron oxide nanocrystals forming in hot solutions of the molecular precursor at longer duration exposures to Ar bubbles might be explained directly by CNT as the result of solution temperature lowering (i.e., an increase in monomer supersaturation), the coincidental PSDsharpening effect also observed can be ascribed to better heat dissipation by the bubbles (i.e., faster system relaxation/lower β value),

Figure 1. Schematic illustration relating the kinetics of a NP-forming process that is nucleation rate-limited, starting from a heated solution of the monomer precursors. The kinetics are controlled via Ea,2, the activation energy of the nucleation step in the process (and not the chemical reaction, whose transition state is denoted as 00 TS00 in defining Ea,1, the activation energy barrier of that step). The distributed nature of the activated state (critical nuclei) energy levels, in relation to Ea,2, is depicted via the stacked lines; the resulting dispersion in the activation energy barrier (which is describable via a t-dependent activation entropy and gives rise to the observable distribution of activation energies) is depicted by the overlaid red curve. As the conversion proceeds, this static energy diagram (essentially a snapshot of the process at a single t) evolves such that the magnitude of Ea,2 (i.e., ΔG2q) effectively decreases over time (while the activation enthalpy, which includes such things as the monomer desolvation energy and the nucleus surface tension at constant geometry, remains fixed) as the number of accessible energy levels in the activated state increases due to the increasing critical nucleus size/entropy. The final NP state closely reflects the distribution of particle sizes/energies observed in the activated state, across all t, assuming minimal growth beyond nucleus size; as per CNT, the growth of nuclei beyond the critical size is energetically favorable, and thus it is not treated as a distinct activated process. The change in the Gibbs free energy, relating the overall process thermodynamics, is given by: ΔG2 ΔG1.

as should become clear in the next section. Provided that the nucleation step is rate limiting, the coordinate diagram pertaining to such a conversion is depicted schematically here in Figure 1. 3. Nucleation Rate-Limited Kinetics. In a recent work,44 the derivation of the dispersive kinetic model, below, that is useful in describing nucleation rate-limited conversion versus time (x t) transients was provided ðβt 2 Þ

x ¼ e½α=t½e

1

ð1Þ

In eq 1, x refers to the relative amount of the reactant phase (e.g., free monomers) remaining in the system at time, t (correspondingly, plots of 1 x vs t relate the product—NP—fraction). The model is particularly useful for modeling 00 acceleratory00 , sigmoidal x t transients, i.e., asymmetrical, S-shaped curves in which the rate of conversion, postinduction period, is faster after the inflection point in the curve than before it; the opposite is true for 00 deceleratory00 kinetics, as will be discussed later. Acceleratory kinetics are typically associated with nucleation rate-limited processes, while deceleratory transients reflect denucleation behavior frequently observed during crystal decomposition or dissolution.42,44,81 As a point of interest, Madras and McCoy82,83 used the assumption of an infinite denucleation rate in their modeling of PSD t-evolution for NPs that can grow/dissolve and aggregate, following their initial 216



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formation via nucleation. Similarly, Robb and Privman62 assumed the irreversible capture of diffusing monomers during the rapid growth of supercritical clusters (i.e., the burst nucleation period, which they assumed to be followed by diffusional growth). Those examples lend support to the use of eq 1 in modeling cases whereby nucleation, alone, is rate-limiting (provided that the nucleation kinetics are indeed dispersive and that the functional form of the dispersion is adequately related by that equation42). The physical significance of the two fit parameters, α and β, in eq 1 has been discussed elsewhere.44 Briefly, α is closely related to a traditional rate constant that is linked to a fixed activation energy potential, Ea0 (which is directly related to the t-independent, activation enthalpy of the conversion). On the other hand, β relates the t-dependent change in the activation entropy of the conversion, and it is responsible for imparting sigmoidal character to otherwise exponential/first-order x t transients that would appear if β = 0 (i.e., in the limit of nondisperisve/classical kinetics). Together, those two fit parameters were recently used to define the probability distribution function, pdf, of activation energies, g(Ea), corresponding42 to eq 1

Figure 2. Theoretical molar PSDs plotted using eq 5, assuming that r0 = 0 and the values of the interfacial tension and monomer volume described in the text. The other relevant physical parameters are as follows: (red) α = 0.1 s, β = 0.002 s2, T = 298 K; (black) α = 0.001 s, β = 0.00002 s2, T = 298 K; (blue) α = 0.001 s, β = 0.0002 s2, T = 298 K; (green) α = 0.001 s, β = 0.0002 s2, T = 498 K; (purple) α = 1 s, β = 0.0002 s2, T = 298 K; (orange) α = 0.001 s, β = 0.2 s2, T = 298 K. The abscissa is plotted from 0 to 50 nm. A lower area distribution, e.g., the green curve, is thought to indicate that the formation of nuclei/NPs is less favorable (i.e., less probable) at that temperature/with those physical parameters than in other cases (e.g., the purple curve traces out a larger relative area corresponding to the production of a larger quantity of NPs). It is known from CNT that nucleation is generally faster at lower T because of higher system supersaturation and the formation of smaller nuclei/NPs occurs via higher activation energies (e.g., lower α values42) than the formation of larger ones. The key difference between the black and orange curves is the slower system relaxation in the latter, as defined by the larger β value (which is positive, see eq 1), leading to greater dispersion/PSD broadening.

) pffiffiffi !( 0 pffiffiffiffiffiffiffiffi ðE 0 E Þ=k T a B α β eðEa Ea Þ 1 kB T ½e a pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðEa 0 Ea Þ kB T Ea 0 Ea qffiffiffiffiffiffiffiffiffiffiffi

gðEa Þ ¼

α

e

0 βkB T ½eððEa Ea Þ=kB TÞ Ea 0 Ea

1

ð2Þ

In eq 2, Ea represents the t-dependent activation energy (corresponding to the Gibbs free energy of activation, which, in turn, represents a combination of the enthalpy of activation and entropy of activation) that is common to all dispersive kinetic processes;3335,44 kB is the Boltzmann constant; and T is the absolute temperature. Plots of eq 2, for different values of the physical parameters contained within it, can be found elsewhere.42 Note that the width of the activation energy distribution would be infinitely narrow if β = 0, as per the case of nondispersive (classical) kinetics where Ea = Ea0. Assuming a spherical particle morphology, the PSD, g(r), can be related to the pdf function in eq 2 by utilizing the Kelvin/ GibbsThomson equation to link the critical nucleus radius, r, directly to the activation energy of nucleation, Ea, as shown below r ¼

2γvm Ea

Plots of eq 5 are shown in Figure 2; in all cases it is assumed that γ = 0.3 J/m2 (neglecting the Tolman length, for simplicity), as per ref 62, and that vm = 600 Å3 (a typical value for a small drug molecule, based on the author’s experience42). Due to the fact that monomers do not have a particle radius, in the nucleation rate-limited conversions simulated in Figure 2 it was assumed that r0 = 0 in all cases. From Figure 2, it is possible to see that all of the PSDs are skewed in the direction of larger particle radii (i.e., to the right). Interestingly, the NP modeling work by Robb and Privman,62 relying on CNT/LaMer’s mechanism to describe the nucleation kinetics and the numerical solution of a master equation for combined irreversible nucleation and diffusional growth, also yielded PSDs that generally tail in the same direction. Such distributions have been attributed62 to the nucleation of very small NPs,47,51,60,84,85 consistent with the derivation of eq 5 presented here (starting from eq 1, which was first proposed six years ago86). Additional compelling support for the PSD shapes generated in Figure 2 comes from the work of Wette et al.,87 who studied the nucleation kinetics of an aqueous suspension of charged colloidal spheres under deionized conditions and similarly observed PSDs skewed slightly toward larger particle sizes (as compared to completely symmetrical, Gaussian distributions). From Figure 2 it is clear that to generate the smallest (and least asymmetrical) NP PSDs using only the nucleation mechanism it is beneficial to use high T. To demonstrate a real-world example, two activation energy distributions previously generated to characterize the nucleation rate-limited conversion of anhydrous crystalline carbamezepine

ð3Þ

where γ is the interfacial tension (uncorrected for the nucleus/ NP surface curvature) and vm is the monomer volume. Equation 3 can be modified to yield Ea Ea 0 ¼

2γvm r r0

ð4Þ

where r0 is an arbitrary starting length for the nucleus radius at t = 0 (to be used where applicable, vide infra). Substituting eq 4 into eq 2 and solving yields the following relationship

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 βðr0 rÞA gðrÞ ¼ @α 2γvm ( ) pffiffiffiffiffiffiffiffi eð2γvm =ðr0 rÞÞ kB T ðr0 rÞ½eð2γvm =kB Tðr0 rÞÞ 1 pffiffiffiffiffiffiffiffi 4γvm kB T pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi βkB Tðr0 rÞ ð2γvm =kB Tðr0 rÞÞ ½e 1 2γvm ð5Þ eα 217



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solid-state kinetic model) to describe the homogeneous nucleation kinetics in their system provides further justification for the notion that nucleation is a dispersive process since, in recent years, Plonka successfully derived the general form of the Avrami equation starting from the dispersive kinetic assumption of fractal conversion times.33,42 As an aside, the Avrami equation has also been reported in recent NP studies,63,78,80 despite its original derivation as a solidstate kinetic model for describing random and isotropic nucleation with simultaneous fixed-rate growth of the resulting spherical particles in the system, up to the point of their impingement (hence the author’s use of the nomenclature 00 nucleation-andgrowth00 for that mechanism). Note also that concentrations are intentionally not used in eq 1 or eq 6 (as per traditional solidstate kinetic models that rely on the fractional conversion metric, 1 x) since nuclei/colloids/NPs/crystals can exhibit a sizedependent reactivity due to their different surface-to-volume ratios (assuming fixed morphology and chemical composition) that cannot be adequately described using only the solutionphase property of molarity. Kinetic models based on traditional (nondispersive) solution-phase mechanisms inherently assume that system homogeneity is maintained throughout (which is generally untrue for phase transformations), and they ignore particle size effects (e.g., one should not assume that a 00 one molar00 suspension of NPs will exhibit the same specific reactivity as large crystals in the same solvent system and at an equivalent 00 concentration00 ). On the other hand, of the traditional solidstate kinetic models found in the literature,45 only the Avrami equation can fit sigmoidal x t transients long known to result from nucleation rate-limited conversions; the other models are useful mainly in characterizing the kinetics of processes that are rate-limited by mechanisms operating on more macroscopic scales (e.g., crystal growth) and that are nondispersive. Under the burst nucleation paradigm of LaMer,1820 the explosive growth of nuclei creates a rapid depletion of the immediate monomer supply. That process creates a separation between the nucleation (denucleation) and growth (dissolution) time scales that is beneficial to certain NP preparations in that it allows NPs to form that are of similar size to the phase formed right after nucleation,7 as can be observed experimentally, e.g., using the familiar synthetic technique of hot injection.2,3,4648 In that regard, it is clear that NPs might serve as more stable (and thus more easily studied) surrogates for nuclei, which are inherently unstable because they represent an activated state (akin to a transition state in solution/gas reaction kinetics) for phase transformation, as per CNT (and that is why nuclei must grow once they form, provided they exceed critical size and monomers are available—vide infra). In a recent work, Shevchenko et al.7 reported that CoPt3 NPs formed via hot synthesis 00 nucleate and grow up to their final size at an early stage of the synthesis with no OR observed upon further heating00 . That finding further supports the idea that it is possible, at least in certain applications, that nucleation is the only mechanism that needs to be considered in describing the NP formation kinetics (via eq 1) and/or NP PSDs (via eq 5); that point made, other mechanisms also warrant further discussion in this work. 4. Denucleation Rate-Limited Kinetics. To describe deceleratory, sigmoidal x t transients, such as those relevant to denucleation44,82 rate-limited processes, the author has found success in using the following dispersive kinetic model44

Figure 3. Molar PSDs corresponding to the 00 real-world00 nucleation ratelimited phase transformation of anhydrous carbamezepine to the hydrated crystal form at 15 °C. The red line depicts the control case without any additive, whereas the black line corresponds to the same conversion occurring in the presence of PEG. The PSDs plotted here using eq 5 correspond to the activation energy distributions plotted in Figure 9 of ref 42. The values α = 8.7 102 min and β = 9.9 104 min2 were used in the plot of the no-additive case, while the values α = 9.0 102 min and β = 6.9 104 min2 were used in the PEG additive case; both parameter combinations were determined elsewhere30 and subsequently used in the earlier modeling work.42

to its hydrated crystal form,42 both in the presence of the nucleation inhibitor polyethylene glycol 6000 (PEG) and without it, were converted to corresponding PSDs using eq 5. The resulting distributions, plotted in Figure 3, show that the inhibiting additive results in a PSD shifted slightly to smaller particle sizes as compared to the no-additive case. That finding is consistent with the notion that PEG slows down the conversion kinetics because smaller nuclei require a greater work of formation (as per CNT). In both cases, the median critical nucleus radius is ∼12 nm, corresponding to a NP volume of ∼7200 nm3 (assuming a spherical geometry), whereby the volume of the triclinic unit cell of the Form I anhydrate is known88 to be ∼2.4 nm3. The predicted PSDs, falling in the nanometer range, are consistent with the mechanism of nucleation (of course, the particles that initially form must grow to their final observed size, typically on the micrometer scale, via some non-rate-limiting crystal growth mechanism). As a point of interest, Wette et al.87 also observed that the nucleation rate density, following a brief induction period, increases sharply, then passes through a maximum (which is often quite flat), and then decreases again, first rapidly and then more slowly. In the author’s opinion, that finding corroborates the idea that nucleation is a dispersive kinetic process in that the observed distribution of nucleation rates (corrected for the remaining free volume in the melt, which is also t-dependent) is likely to be supportive of an underlying distribution of activation energies. Furthermore, the work appears to support the author’s use of a Gaussian approximation for the t-dependence of the rate coefficient, which was originally used to derive the dispersive kinetic models presented herein (eq 1 and eq 6),44 noting that an approximate Gaussian fit to the data of Wette et al.87 was also recently performed by Shields et al.63 for use in their modeling. Lastly, the use of the JohnsonMehlAvramiErofe’ev Kolmogorov (JMAEK or 00 Avrami00 26) equation by Wette et al.87 (more specifically, their own discrete adaptation of that classical

ð βt 2 Þ

x ¼ e½αt½e 218

1

ð6Þ



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where the rate parameters, α and β, are analogous44 to those in eq 1, noting that β is explicitly shown to be negative in eq 6 (thus, curve fits with this equation will return positive values) because it reflects the deceleratory trend in the empirical data (whereby the conversion rate deceleration, as a function of t, is due to a loss in activation entropy; the corresponding increase in the effective activation energy barrier resultantly slows down the conversion rate). In both nucleation and denucleation cases, the system relaxation becomes faster as β decreases. For denucleation processes, the pdf of the activation energies, g(Ea), is related by the equation shown below42 gðEa Þ ¼

!(pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 0 0 Ea 0 Ea ½eðEa Ea =kB TÞ ½eðEa Ea =kB TÞ 1 α pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kB T βkB T 2 Ea 0 Ea

qffiffiffiffiffiffiffiffiffiffiffi

α

e

Ea 0 Ea ðEa Ea 0 =kB TÞ ½e βkB T

1

ð7Þ

Figure 4. Theoretical molar PSDs plotted using eq 8 and the values for interfacial tension and monomer volume as per Figure 2. In all cases, it was assumed that r0 = 100 nm. The other relevant physical parameters are: (red) α = 0.001 s1, β = 0.0002 s2, T = 298 K; (black) α = 0.001 s1, β = 0.002 s2, T = 298 K; (blue) α = 0.0015 s1, β = 0.002 s2, T = 298 K; (green) α = 0.001 s1, β = 0.0002 s2, T = 498 K; (purple) α = 0.0001 s1, β = 0.0002 s2, T = 298 K; (orange) α = 0.001 s1, β = 0.2 s2, T = 298 K. The abscissa is plotted from 0 to 90 nm. Note that the finite g(r) values at r = 0 correspond to nonzero monomer populations of completely dissolved atoms/molecules; additionally, the key difference between the red and orange curves is the faster system relaxation in the latter, as defined by the lower β value (which is negative; see eq 6), leading to less dispersion/a narrower PSD. As per Figure 2 and Figure 3, the distributions are not normalized.

Plots of eq 7 for different values of the various physical parameters contained within it can be found elsewhere.42 Plots of the corresponding NP PSDs, g(r), related to those activation energy distributions were obtained using eq 8 (below) and are shown in Figure 4. In all cases, it was assumed (arbitrarily) that r0 = 100 nm, for modeling purposes.

! α gðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi βkB T 9 8 2γvm 2γvm > > Þ > > ð Þ pffiffiffiffiffiffiffiffiffiffiffi ð > > p ffiffiffiffiffiffiffiffiffiffi < 2γv ½e kB Tðr r0 Þ k Tðr r Þ B 0 1= r0 r ½e m pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > > 2 2γvm kB T r0 r > > > > ; : e

α

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2γvm βkB Tðr0 rÞ

2γvm ð Þ ½e kB Tðr r0 Þ

1

nucleation, the two dispersive mechanisms might, under conditions of good rate balance, produce narrower and more symmetrical PSDs than is possible via either mechanism, separately. Such dual-mechanism processes are discussed more in the next section, while OR is described further in Section 7. 5. Bimodal NP Distributions and PSD Focusing: Combining Mechanisms for Simultaneous Nucleation and Denucleation. Reports of bimodal PSDs3,54,55 and PSD focusing/ defocusing3,21,46 can be found in the recent literature. Of most interest to experimentalists is PSD focusing93 because it results in sharper, more symmetrical (Gaussian) distributions. Focusing behavior was successfully predicted by the stochastic, numerical modeling approaches of Talapin et al.46 and van Embden et al.48 to occur under conditions relevant to the hot injection synthetic technique by producing high monomer supersaturations that favor burst nucleation (presumably followed by OR at later times, provided that the burst is not instantaneous and does not yield completely symmetrical PSDs, as should become clear later). Alternatively, Sugimoto21 suggested that a balance between monomer production (from reaction of the precursor species) and consumption (by NP formation/evolution) could maintain the supersaturation at a steady-state level and, consequently, lead to PSD focusing/monodispersity. While modeling of PSD focusing and defocusing phenomena (via dynamic PSD evolutions) has been performed, the origin of bimodal PSDs has largely been ignored with only a fundamental thermodynamic perspective being presented in the recent literature.54 Fortunately, using the g(r) functions already discussed in this work, it might be possible to explain both PSD-related phenomena very simply. As can be inferred from the last section, certain bimodal NP PSDs can be comprised of two elementary PSDs originating from either distinct (i.e., having different Ea0 values) nucleation events, combined nucleation and denucleation events, or two coupled

ð8Þ

From Figure 4 it can be seen that the PSDs predicted by eq 8 are all skewed in the opposite direction to those in Figure 2. While such PSDs have been reported in the NP literature,3,89 the explanation that this kind of PSD shape might be caused by a single, rate-limiting (dispersive) denucleation step is new. Using the basic premise of dispersive kinetics, one cannot simply assume that all NPs will form precisely at the same time, even when only a single mechanism is rate-limiting; instead, one should expect to observe the effect of a distribution of activation energies, leading to, e.g., a 00 spread in the nucleation time00 .55 The PSD asymmetry observed thus far, corresponding to either eq 5 or eq 8, can be corrected by size distribution focusing, a phenomenon that has also been widely observed experimentally. PSD focusing, which is discussed more in the next section, might occur as the result of 00 smaller particles catching up in size to the larger ones because the growth rate of nanoparticles decreases as the size increases00 90 (whereby only the second part of the statement is true for OR, vide infra). Alternatively, one can consider that smaller particles, being inherently less stable than the larger ones, predominantly dissolve into their constituent monomers and, resultantly, allow their larger counterparts to grow (perhaps by inducing localized concentration gradients,91 of the type that can sometimes produce oscillations via the phase transformation step of certain processes92). That process, which is known to be important during OR, is analogous to denucleation. Denucleation, by itself, forms NP PSDs of the variety described by eq 8 and graphically depicted in Figure 4. However, when combined with 219



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course, that situation corresponds to widely observed PSD focusing behavior. Two scenarios that can give rise to NP PSD focusing (and defocusing) at different t are presented in Figure 5. To account for PSDs that are observed as the result of two simultaneous dispersive kinetic processes, one can use a mathematical convolution of the individual PSDs. The distributions can be convoluted in the following manner: ga*ga, gd*gd, and ga*gd (which is the same as gd*ga, because of commutativity), where ga is a unimodal PSD originating from a nucleation mechanism and gd from denucleation; the convolution operation is defined for two functions, g1 and g2, as shown below ðg1 g2 ÞðrÞ ¼ ¼

Z ∞

∞

Z ∞

∞

g1 ðFÞg2 ðr FÞdF g2 ðFÞg1 ðr FÞdF

ð9Þ

Using eq 9, one can envision how PSDs based on eq 5 and eq 8 (for ga and gd, respectively), which have asymmetric tails pointing in opposite directions, can together (when convoluted) produce a more symmetrical, Gaussian-looking PSD as was suggested in the previous section.95 6. Few Remarks on ParticleParticle Interactions, Aggregation, and Coalescence. While NP formation has been attributed to nucleation-and-growth,7 burst nucleation followed by diffusional growth,62 and simultaneous nucleation, growth/ dissolution, and coarsening (OR) mechanisms,39,46 the fusion, aggregation, and/or oriented attachment of smaller particles have also been presented as viable NP-based mechanisms in the literature.50,63,96104 There is recent evidence of particle particle interactions impacting the PSD evolution of NPs, as observed by TEM (whereby those interactions were found to play a key role in OR).54 Such interactions have also been supported via direct observation of individual NP growth kinetics.55,105 While there is literature evidence to support the existence of an aggregation/oriented attachment/coalescencetype NP growth mechanism (and such a mechanism has even been included in some numerical simulations of PSD evolution82,99), from a recent paper by Zheng et al.55 it can be seen that different NPs—even those in the same system—when monitored separately can evolve via different mechanisms (and thus exhibit different growth kinetics). That is to be expected since, at any given time during conversion, the elementary mechanisms of nucleation, denucleation, diffusional growth, aggregation, etc., can all potentially occur simultaneously (as per the previous section of this paper), with at least some degree of overlap in time scales. Even OR (discussed more in the next section), alone, can give rise to complexities provided that the NP kinetics are considered on an individual basis.106 The addition of aggregation/oriented attachment/particle coalescence (which are not treated in this work) to the mechanisms already discussed can potentially further complicate the treatment of NP process-related kinetics. To simplify the problem, it is of paramount importance to determine the rate-limiting mechanism of the major NP fraction, as a function of t; the PSDs discussed thus far should provide a means for doing that, albeit for only two mechanisms. In the next section, the stationary PSDs related to OR will be discussed briefly to add this well-established NP mechanism to the candidate list. 7. LSW Theory for Treating OR. OR refers to a kind of particle coarsening107 that results in the growth of larger NPs at the

Figure 5. Schematic representations (not drawn to scale) of the time evolution of the mean particle radius, under isothermal conditions, via two different processes whereby: (A) the conversion is denucleation (red) rate-limited in the beginning (note: this does not imply that NPs did not form at the outset of the conversion, e.g., via burst nucleation, only that the nucleation mechanism is not rate-limiting at early t), then it transitions to a nucleation rate-limited conversion at t = t1 (blue) and, eventually, to an OR rate-limited conversion at longer times (at t = t2; green) and (B) the conversion is nucleation rate-limited in the beginning (blue), then it transitions to a denucleation rate-limited conversion at t = t1 (red) and, eventually, to an OR rate-limited mechanism (at t = t2; green). In scenario (A), PSD focusing occurs in the regions around both t1 and t2 due to the fact that the adjacent mechanisms operating there produce NP PSDs that tail in opposite directions. Around t1, the focused PSDs also correspond to the smallest possible (stable) NP sizes that can be formed; however, that is not the case if the conversion is allowed to proceed for longer conversion times (i.e., with PSD focusing in the vicinity of t2). In scenario (B), PSD focusing occurs only in the region around t1; that is also the region where larger NPs can be expected to form (provided that t , t2). For real-world syntheses, perhaps the most common mechanism, nucleation-and-growth followed by OR, is reflected by the last two curves in (A).

denucleation rate-limited processes—depending on the relative shapes of the two constituent distributions. Along similar lines, Chen et al.94 previously described bimodal PSDs by combining two unimodal populations. Taking that approach a step further, PSD focusing might result as a natural consequence of two mechanisms (e.g., nucleation and denucleation) being simultaneously rate-limiting (in cases where the NP formation/ dissolution is a quasi-equilibrium process).46,90 For example, while the PSD might exhibit a shape similar to that in Figure 4 at earlier t, the result of a predominantly denucleation rate-limited process, at later t the PSD tail might become reduced, disappear completely, or eventually point in the opposite direction (toward larger r values) as a secondary nucleation rate-limited mechanism begins to participate or even dominate the process. In cases where a PSD tail essentially disappears entirely over a given period of time, yielding a narrow and symmetrical PSD, the two competing mechanisms are well-balanced in terms of their opposing rates (ultimately, giving rise to a similar critical nucleus size for both nucleation and denucleation, at relevant t); of 220



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larger than monomers112 (so-called 00 clusters00 ), rather than particle coarsening/OR. Regardless, while OR describes particle size evolution over time, it is not a conversion mechanism in the same sense as nucleation and denucleation (only the latter two mechanisms represent phase transformations); that stated, its role in the evolution of homochirality in Viedma conversions cannot be discounted solely on that basis. As an alternative to modeling real-world PSDs, using eqs 5, 8, 10, or 11, one can more readily characterize the t-evolution of the mean particle radius; in the case of diffusion-controlled OR, the cube of , corrected for the same power of the starting particle radius size, evolves linearly as a function of t, while the same is true for the square of in reaction-controlled OR cases. It is to be expected that should evolve quite differently for the two mechanisms giving rise to the PSDs in eq 5 and eq 8, and as compared to OR, as will be discussed in the next section. Note also that while it has been discussed already that bimodal PSDs can be produced via the combination of (at minimum) two NP populations formed by different mechanisms, or the same mechanism operating on distinct time scales, clearly OR can potentially serve as one of those mechanisms (however, OR alone does not generate bimodal PSDs).62,63 Similarly, OR is likely also to be important in PSD focusing.46 A recent work by Polte et al.60 showed the particle focusing effect in the preparation of gold NPs, whereby the PSDs observed early in the conversion (generated via the reduction of Au3+ in the starting material complex) exhibited a distinct right-hand skew. Those distributions became more symmetrical (Gaussian) as the conversion, tracked via the gold oxidation state, proceeded to completion. A possible explanation for that behavior, in lieu of the more complex, four-step mechanism proposed therein, could be that a nucleation rate-limited mechanism dominates during the early stages of the conversion (producing PSDs of the variety depicted in Figure 2) and then transitions to a particle coarsening mechanism at longer t (whereby the PSDs would ultimately become left-skewed as t f ∞). As discussed in Section 5, the PSD asymmetries resulting from different mechanisms can largely cancel each other out over finite t intervals, leading to the observation of more symmetrical/narrower PSDs during the process. Once again, relevant examples giving rise to PSD focusing are depicted schematically in Figure 5. Similarly, the monodisperse thiol end-capped Au NP PSDs ultimately obtained by the synthetic procedure of Jin et al.114 might also reflect a nucleation rate-limited mechanism (after reduction of the Au+ intermediate) that produces an initially polydisperse/broad NP distribution, followed later by OR that gradually sharpens the PSD via heating over ∼24 h in the presence of excess thiol, that is, provided that the key factor influencing NP stability (and, consequently, leading to the 00 survival of the robustest00 114 solid-state species) is size, as conveyed via the Kelvin equation in the case of OR, and it does not involve other variables (e.g., particle geometry).114 8. Time Evolution of the Critical Nucleus during Nucleation. The PSDs described by eqs 5, 8, 10, and 11 are all stationary/steady-state functions since they are independent of t. However, that is not to say that the particle size does not evolve as a function of time, under each of the corresponding ratelimiting mechanisms. With regard to the case of nucleation ratelimited conversions (see Section 3), the equation, below, describes the t-evolution of the mean critical nucleus radius

expense of (i.e., accompanied by the dissolution of) smaller, less thermodynamically stable ones. OR generally occurs on a longer time scale than nucleation and/or NP growth;62,82 for that reason, it is often possible to separate nucleation (denucleation) and growth (dissolution) rate-limited processes from OR, in time, analogous to the way in which the LaMer mechanism of burst nucleation can be separated from subsequent NP growth. OR can be described using classical LifshitzSlyozovWagner (LSW) theory107109 under conditions of low supersaturation and quasi-equilibrium. Of primary interest here, LSW theory predicts limiting/stationary NP PSDs for both reaction-controlled and diffusion-controlled growth regimes, analogous to the author’s dispersive kinetic models. The t-evolution of NP PSDs via OR has been modeled directly via Monte Carlo simulation46 and by numerical solution of population-balance differential equations;48 however, e.g., the assumption of an initial Gaussian distribution of nuclei/NPs at the start of the simulation is somewhat dubious in light of the findings presented in Section 3. In the diffusion rate-limited case of OR, it is known from LSW theory that ! 34 e 2 ð1=ð1 2u=3ÞÞ ue 25=3 ð10Þ gLSW ðuÞ ¼ 11=3 if 0 < u < 1:5 7=3 3 u ðu þ 3Þ 2 gLSW ðuÞ ¼ 0 otherwise where u r/ and is the average particle radius. Similarly, in the case of reaction rate-limited OR gLSW ðuÞ ¼

27 3u ð3u=ð2 uÞÞ e if 0 < u < 2 ð2 uÞ5

ð11Þ

gLSW ðuÞ ¼ 0 otherwise Plots of eq 10 and eq 11 can be found elsewhere.46 Additionally, the t-evolution of the mean particle radius and of the PSD has been plotted in other works, for both types of OR.46,48 In general, the NP PSDs produced by coarsening are most similar to those presented in Figure 4, relevant to denucleation, in that they are both skewed to the left. That similarity can be traced back to the fact that the solubility of the solid is a determinant factor in OR kinetics, as it is directly related to the validity of LSW theory.110 That stated, it might be difficult to qualitatively differentiate PSDs formed via denucleation vs coarsening ratelimited mechanisms without either a more detailed knowledge of the procedure used in a given NP preparation and/or knowledge of the time scales involved. Fortunately, in the case of chiral symmetry breaking via the Viedma process,111 which involves a combination of several different elementary mechanisms (via coupled equilibria), the idea that the key mechanism is more likely to be nucleation rate-limited112 than ripening/coarsening rate-limited113 should be verifiable simply by inspection of the PSDs (monitored down to the nanometer scale) generated during the course of the conversion—as discussed above, coarsening gives rise to PSD shapes that more closely resemble those of denucleation than nucleation, making at least those two mechanisms relatively easy to distinguish. Perhaps, however, the proposed mechanism termed 00 Viedma ripening00 113 might actually be akin to secondary nucleation occurring via subunits

¼ 221

2γvm Ea kB Tβt 2 0

ð12Þ



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conversion, the smallest critical nuclei with the highest activation energies are first to form. That observation is consistent with CNT because smaller nuclei are known to form at higher supersaturations that are typically present at the outset of a nucleation event. The larger critical nuclei that form at longer t produce greater dispersion in the Ea distribution, resulting in a lowering of the overall activation energy barrier. From a monomer-level perspective, it has been discussed previously42 that, during nucleation, the larger nuclei that form exert an increasingly strong attractive force on the surrounding monomers. It is that force that increases the rate of monomer motion in the vicinity of each nucleus/NP, as a function of t. Interestingly, 00 an additional force between two masses which increases linearly with the distance of the masses00 has been reported in the recent astrophysics literature for application to galaxies in clusters; that force was used in a mathematical description of an 00 ever oscillating universe that underwent a geometrical phase transition in some distant past00 ,115 whereby the potential about the transition state can be adequately represented by an inverted SHO-like parabola, consistent with the author’s previous paper42 (in which the reason that the activation energy barrier maximum was designated to be spatially located at the nucleus interface should become clearer in a future work). In the complementary case of denucleation, the dispersive kinetic model indicates that the first to form are the largest critical nuclei that have the lowest activation energy barrier (work of formation), consistent with the CNT notion that at the outset of crystal dissolution one can expect a very low monomer concentration in solution. As the conversion proceeds, the critical nuclei become smaller (as the solution concentration approaches saturation), coinciding with an increases Ea (decrease in activation entropy). That greater activation barrier slows down the overall rate of the conversion, imparting the characteristic deceleratory shape to experimentally observed x t transients.42,44 As for the microscopic view of the process, one can expect anomalously slower (non-Brownian) monomer diffusion away from the surface as the critical nucleus size gets increasingly smaller, as a function of t. In any event, nucleation and denucleation clearly produce very different mean particle size evolution behavior than OR (e.g., see Figure 5), thus making all three mechanisms readily distinguishable from each other via simply monitoring as a function of t.

Figure 6. Theoretical plots of the time evolution of the mean NP radius during a nucleation rate-limited conversion, obtained using eq 12. The relevant physical parameters used in the modeling are as follows: (red) Ea0 = 85 kJ/mol (see ref 42), vm = 600 Å3, γ = 0.3 J/m2, β = 0.002 s2, T = 298 K; (black) Ea0 = 85 kJ/mol, vm = 600 Å3, γ = 0.3 J/m2, β = 0.0008 s2, T = 298 K. The value of Ea0 corresponds to r0 ∼ 2.5 nm in each case (see ordinate intercept). The conversion must reach completion at some finite conversion time, t, since r f ∞ as Ea f 0, yet the monomers will eventually become depleted by the increasingly rapid specific conversion rate. Regardless, it is likely that another mechanism, e.g., OR, begins to dominate at longer t (see Figure 5 and text for details).

Equation 12 was obtained using eq 3 together with the description of the t-dependent Ea for dispersive nucleation that can be found elsewhere.42 The complementary result for denucleation (see Section 4) can be derived analogously (not shown here). Figure 6 shows representative plots of the critical nucleus radius, as it evolves with time, during a given nucleation ratelimited phase transformation. Clearly, increases as a function of t, as one might expect based on experimental evidence. For example, in ref 55 (Figure 4B, data set 00 e00 ) a single NP growth trend is observed that appears to support eq 12 in this work, suggesting that the formation of that NP was nucleation ratelimited. In the modeling approaches of Robb and Privman,62 Ribeiro et al.,101 and van Embden et al.,48 the critical nucleus size was similarly observed to increase rapidly over shorter t (i.e., during the initial nuclei burst), followed by a linear growth regime at longer times (that type of particle size evolution has also been observed experimentally for iron oxide nanoparticles3). The transition from rapid critical cluster growth to linear growth might reflect a change in the mechanism, as discussed above for the case of OR. Along those lines, since eq 12, by itself, produces a “catastrophe” at long t, such a mechanistic change would be welcome from a physical standpointe.g., see Figure 5. Equation 12 relates an expected critical nucleus/NP size increase during a given nucleation event, e.g., as the supersaturation is relieved under isothermal conditions. Thus, 00 nucleation-andgrowth00 seems an appropriate terminology for both the mechanism related by the author’s dispersive kinetic model and the Avrami model (vide supra). The acceleration in the NP radius growth is physically describable by an increase in the activation entropy42 as a function of t that simultaneously lowers Ea and creates the observed rate acceleration in the sigmoidal x t transients; as depicted schematically in Figure 1, the critical nuclei that form as precursors to the final NP PSD have broadly distributed energies, compared to the free monomers in solution and relative to kBT. At the start of the

’ CONCLUSIONS A key challenge in the preparation of NPs lies in obtaining narrow, unimodal PSDs centered about the desired (mean) target size. That is because the properties of NPs can change dramatically with size. Understanding NP formation, growth and long-term evolution is paramount in being able to tailor/ control PSDs via advances in synthetic strategies.116 As stated by Owen et al.,117 00 despite ready access to a variety of colloidal nanostructures, these crystallizations remain difficult to control because the mechanisms of their growth are not well understood00 . Complicating their kinetic analysis is the fact that there are different types of synthetic approaches that utilize different experimental conditions/procedures/precursors, thereby potentially favoring different mechanisms. This work demonstrated that one can qualitatively assign the mechanisms of nucleation, denucleation, and/or OR to experimentally observed PSDs (at steady-state; as t f ∞) based on their different shapes and the time scales over which they operate. Furthermore, the 222


The Journal of Physical Chemistry C observation of bimodal PSDs and the phenomenon of PSD focusing were revealed to originate from a combination or convolution of two or more of those mechanisms. Lastly, different kinetic transients for the time evolution of the mean NP radius size can also be used to distinguish between them. An important possibility not discussed herein is the case where the reaction of (different118) solute precursors, leading to the formation of free/00 active00 monomers in solution (see Figure 1), is rate-limiting.117 In such cases, the conversion kinetics are determined by solution phase reaction mechanism(s) and not the solid-state mechanism(s) discussed in this work. Regardless, when considering limiting NP PSD shapes and/or the mean NP radius time evolution, i.e., observable properties of solids, appropriate data treatments should nonetheless employ solid-state models and not solution-based ones. For that reason, the practice of using so-called 00 reporter reactions00 , which relate only the rate of net monomer production/consumption in the solution phase, to describe the kinetics of processes taking place in the solid state (e.g., where OR can be rate-limiting) should be applied judiciously. A better experimental approach might be to utilize simultaneous measurements of both solution-phase and solidstate species, to help pinpoint the rate-limiting mechanism at any given time and/or to demonstrate a correlation between the kinetics in both phases. While recent modeling efforts have provided good insights into NP formation and PSD time evolution, those approaches are generally more sophisticated than the one presented herein, making them too cumbersome for routine use by nonspecialists— not to mention that the same sophistication (in treating multiple mechanisms, simultaneously, utilizing a variety of empirical inputs) that affords them greater flexibility in reproducing empirical observations also limits their generalization/predictive power. Furthermore, classical descriptions of the nucleation (denucleation) kinetics, having known limitations, are typically retained in many treatments, and while the asymmetric PSDs generated via numerical methods are generally skewed toward smaller particle sizes (to the left), the opposite case and observations of bimodal PSDs (both of which are reported in the experimental literature) have gone largely untreated. In this work, stationary PSD functions for both types of NP distributions were defined using dispersive kinetic models for nucleation and denucleation. By combining the PSDs predicted by each elementary dispersive kinetic mechanism, both together and with the known stationary PSDs predicted for OR, the observation of bimodal PSDs and the phenomenon of PSD focusing were rationalized. Lastly, it is noted here that while the PSDs generated by OR are qualitatively similar to those produced via denucleation (although the time scales of the two processes are often different) the mechanisms can be easily distinguished from each other via their different mathematical descriptions of the time evolution of the mean particle radius, as shown in Figure 5. The theoretical NP PSD functions presented herein, eq 5 and eq 8, are closed-form analytical solutions that rely only on the values of the two kinetic parameters, α and β, to define the PSD shapes (under isothermal conditions). That is because, in dispersive kinetics, the influence of time (whereby past, present, and future conversion times are treated holistically, in a deterministic manner) is inherently incorporated into each PSD via the mathematical relationships that exist between the (t-dependent) critical nucleus size, (t-dependent) activation energy, the Ea distribution, and the overall/observable conversion time. CNT, coupled with dispersive kinetics, indicates that the work of

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formation of the critical nucleus changes over the extent of conversion (e.g., as supersaturation is relieved). That is why increases with t, as per eq 12, during a nucleation rate-limited phase transformation, coinciding with a decrease in the activation energy of conversion that results from increasing activation entropy.44 Activation entropy is a physical quantity that has largely eluded workers investigating nucleation kinetics via computer simulation;32,119 fortunately, recent work by other workers120 has supported its important role in nucleation events.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]; [email protected].

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