Roles of Nucleation, Denucleation, Coarsening, and Aggregation

Article pubs.acs.org/Langmuir

Roles of Nucleation, Denucleation, Coarsening, and Aggregation Kinetics in Nanoparticle Preparations and Neurological Disease Peter J. Skrdla 640 Maple Street, Westfield, New Jersey 07090, United States S Supporting Information *

ABSTRACT: Kinetic models for nucleation, denucleation, Ostwald ripening (OR), and nanoparticle (NP) aggregation are presented and discussed from a physicochemical standpoint, in terms of their role in current NP preparations. Each of the four solid-state mechanisms discussed predict a distinct time dependence for the evolution of the mean particle radius over time. Additionally, they each predict visually different particle size distributions (PSDs) under limiting steady-state (time-independent) conditions. While nucleation and denucleation represent phase transformation mechanisms, OR and NP aggregation do not. Thus, when modeling solid-state kinetics relevant to NP processing, either the time evolution of the mean particle radius or the fractional conversion data should be fit using appropriate models (discussed herein), without confusing/ combining the two classes of models. Experimental data taken from the recent literature are used to demonstrate the usefulness of the models in real-world applications. Specifically, the following examples are discussed: the preparation of bismuth NPs, the synthesis of copper indium sulfide nanocrystals, and the aggregation of neurological proteins. Because the last process is found to obey reaction-limited colloid aggregation (RLCA) kinetics, potential connections between protein aggregation rates, the onset of neurological disease, and population lifespan dynamics are suggested by drawing a parallel between RLCA kinetics and Gompertz kinetics. The physical chemistry underpinning NP aggregation is investigated, and a detailed definition of the rate constant of aggregation, ka, is put forth that provides insight into the origin of the activation energy barrier of aggregation. For the two nanocrystal preparations investigated, the initial kinetics are found to be well-described by the author’s dispersive kinetic model for nucleation-and-growth, while the late-stage NP size evolution is dominated by OR. At intermediate times, it is thought that the two mechanisms both contribute to the NP growth, resulting in PSD focusing as discussed in a previous work [Skrdla, P. J. J. Phys. Chem. C 2012, 116, 214−225]. On the basis of these two mechanisms, a synthetic procedure for obtaining monodisperse NP PSDs, of small and/or systematically targeted mean sizes, is proposed.

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INTRODUCTION Advances in nanoparticle (NP) synthetic procedures, which can provide control over NP size, morphology, and monodispersity, are driven by improvements in kinetic and thermodynamic understanding of the underlying mechanisms. Recent publications in the experimental literature have shown that even in the case of gold NPs, one of the most studied nanomaterials, there is a lack of clarity in the elementary mechanisms that can dominate over the course of their preparation under different experimental conditions. For example, Ji et al.1 proposed both a two-step mechanism (rapid nucleation and slow growth by monomer diffusion, occurring at pH > 6.5) and a three-step mechanism (with intraparticle ripening as the last step, at pH < 6.5) for Au NP formation. Pong et al.2 suggested a mechanism involving (fast) reduction of the Au molecular precursor, followed by formation of small clusters interconnected by amorphous Au, and, last, the growth of the chains up to the point of their collapse (at which point the Au NPs have a radius of ∼7.5 nm). Kimling et al.3 proposed a mechanism involving cluster formation (nucleation) via the Au atoms formed by precursor reduction, followed by agglomeration and compac© 2012 American Chemical Society

tion of the primary particles into polycrystalline NPs with some additional aggregation continuing over longer time periods. Polte et al.4 put forth a mechanism for Au NP formation by classical citrate synthesis that is comprised of four consecutive steps: nucleation, growth of the resulting clusters by aggregation/coalescence, slowing of further growth via diffusion, and, finally, a rapid growth step achieving the final size (∼7.7 nm) that is determined at the point of complete consumption of the molecular precursor. Only during the last step is all of the Au converted to the zero oxidation state; additionally, it is noted here that only the last three steps of that mechanism were considered to be experimentally observable. Using rapid small-angle X-ray scattering (SAXS) and NaBH4 as the reducing agent, Polte et al.5 discussed that the Au NPs form via a reduction and nucleation step very early in the process (200 ms). Along Received: December 20, 2011 Revised: February 6, 2012 Published: February 10, 2012 4842

dx.doi.org/10.1021/la205034u | Langmuir 2012, 28, 4842−4857

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similar lines, Shields et al.6 proposed an “aggregative nucleation and growth” mechanism (described using a modified JMAEK kinetic model; see below), finding that Ostwald ripening (OR) significantly coarsens the NPs only after the more active growth period that they investigated. On the other hand, using a slow reduction/NP forming process (relative to the Brust two-phase synthesis,7,8 which utilizes the stronger NaBH4 reducing agent) to facilitate their time-resolved studies, Sardar and ShumakerParry9 described a mechanism for the formation of nearly monodisperse Au NPs involving both nucleation and NP growth; the highly single-crystalline NPs observed throughout the synthesis in their work were found to be consistent with a diffusion-controlled OR mechanism. Finally, Wu et al.10 used mass spectrometry to discover a size-focusing approach for the preparation of 19-Au-atom-containing nanoclusters, similarly utilizing a weaker reducing agent to establish kinetic control over the nucleation event leading to the initial cluster size distribution, which is followed by growth/aging to the final, stable nanocluster distribution under thermodynamic control (e.g., via OR). Modeling efforts over recent years have also yielded different perspectives on NP formation and growth kinetics. The models/simulations can be subdivided into various classes as per the paper by Rempel et al.,11 whose own numerical simulations of nucleation and NP growth processes led to the identification of five distinct regions: the generation of monomers, small cluster formation, size distribution focusing due to precursor depletion, a quasi-steady-state region and, finally, NP size distribution broadening. The numerical simulations of nucleation and NP growth put forth by van Embden et al.12 similarly showed a quasi-equilibrium period at intermediate times between a rapid growth period observed at the outset of the conversion and the final OR stage of the process that evolves the observed mean particle radius size, albeit largely during the particle size distribution (PSD) focusing event and not so much after it (as per the work of Rempel et al.11). Regardless, both works showed that PSD focusing could occur under reaction or diffusion control, contrary to popular belief at the time (year 2009) that only the latter mechanism was responsible for the phenomenon. On the other hand, the prior numerical modeling work by Talapin et al.13,14 was also able to describe PSD focusing and subsequent defocusing, arguably using only the mechanisms of nucleation-and-growth followed by OR (as per a more recent work15), thus neglecting the establishment of a quasi-steady-state kinetic period. Because the monomer supersaturation affects both of those mechanisms,16 since each fundamentally relies on the Kelvin/Gibbs−Thomson equation for determining the critical particle size needed for continued growth, the resulting competition for monomers can impact both the observed PSD shape and the time-evolution of the mean particle radius size, as should become clear from this work. Clearly, the “unrealistic”12/large supersaturations employed in the previous OR study,13 which led to the most pronounced PSD focusing, are likely to simultaneously induce nucleation. Unfortunately, earlier Monte Carlo simulations13 used generic PSD shapes (symmetrical Gaussian or asymmetrical log-normal distributions) as the starting distributions for OR that may not adequately reflect the PSD shape expected for real-world nucleation events. Likewise, the simulations of Robb and Privman17 assumed a priori either a thermal distribution for subcritical clusters or a normal distribution for supercritical clusters. The population balance modeling approach of Mantzaris18 also did not provide insight into the early PSDs

generated during the nucleation stage. That is why, in this work, steady-state (time-invariant) PSD shapes for both nucleation and denucleation (the complementary process to nucleation) mechanisms will be clarified, comparing them to the well-known limiting PSDs predicted for OR and for NP aggregation rate-limited processes. Despite the known limitations of earlier modeling approaches and the trade-offs generated via their different relative complexities, a class of models not discussed by Rempel et al.11 consists of dispersive kinetic ones.19−21 Dispersive kinetic models are underpinned by the a priori assumption of a distribution of activation energies that imparts a time dependence (i.e., a dynamical component) to the specific rate that is treated as a rate constant by classical kinetic theories. With use of such simple, deterministic, and easy-to-use models, the stationary PSDs corresponding to nucleation and denucleation processes were presented in a recent work.22 Those PSDs can be combined (convoluted), together and/or with the steady-state PSDs predicted for OR, to qualitatively describe the phenomena of PSD focusing and defocusing.22 It was shown22 that size distribution focusing under typical synthetic conditions is likely to require a minimum of two complementary mechanisms that individually produce stationary PSDs that tail in opposite directions (e.g., nucleation and OR; see below). This work is put forth in an attempt to clarify key mechanisms and their associated models that are known to be relevant to the kinetics of NP formation, evolution (growth), and aggregation. Furthermore, the appropriate usage of the models, which are applied to experimental data taken from the recent literature in a variety of fields of intense study, will be discussed. Lastly, the origin of the activation energy of aggregation is described in the context of a newly proposed “orbital theory” that might be more relevant to (and, hence, more insightful into) such systems than classical kinetic theories of chemical reaction rates. 1.1. Nucleation. Nucleation is widely understood to be the first solid-state mechanistic step in NP formation. It involves free monomers coming together to form a critical-sized nucleus of the new, condensed phase. From classical nucleation theory23−27 (CNT) it is known that embryos having a radius smaller than the critical size are unstable and hence they dissociate (i.e., denucleate or dissolve). On the other hand, nuclei/NPs larger than the critical size will continue to grow under thermodynamic control, decreasing their energetic liability corresponding to their surface-to-volume ratio. Because the critical nucleus represents an activated state (akin to the transition state in molecular reaction kinetics, making it is inherently unstable), its growth beyond the critical size is necessary to achieve thermodynamic stability. However, since NP growth is intentionally minimized in most current synthetic approaches (e.g., hot injection that gives rise to burst nucleation) to achieve small particle sizes, the initially observed NP size distributions can be expected to be closely related to the distributions of the nuclei from which they form.22 In that regard, NPs might serve as more stable (and, hence, more easily studied) surrogates for solid-state nuclei. Despite the known limitations of CNT in accurately predicting nucleation rates,29−32 the theory provides important physicochemical insight into the origin of the (steady-state) activation barrier for nucleation and it can be used to predict the formation of monodisperse NP PSDs under steady-state conditions. Recently, the author applied the CNT link between 4843


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the system dynamics do not affect the rate of conversion, the distribution of activation energies collapses into a Dirac δ function, i.e., a single-valued activation energy, as per classical kinetic models. As discussed in recent works,22,29 nucleation is now widely accepted to be a dispersive kinetic process, although capturing that behavior via computer simulation has proven to be quite challenging.32 Therein lays the value of the author’s dispersive kinetic model for nucleation, which has a closed-form analytical solution that is easily regression-fit to real-world experimental data. It provides a coarse-grained description of the process that captures the essence of both the kinetics and dynamical evolution of the system. From refs 22 and 28, nucleation ratelimited conversion kinetics can be related via the following model:

the critical radius size and activation energy, provided via the Kelvin equation, to his dispersive kinetic treatment of nucleation and denucleation mechanisms21 in order to convert the activation energy distributions predicted by the conversion models28 to corresponding PSDs22 reflective of kinetic behavior under usually more realistic, non-steady-state conditions (whereby the critical radius size/activation energy evolves over time as the system relaxes, i.e., as supersaturation is relieved or, conversely, as the monomer concentration builds up in solution in the case of denucleation). Paralleling developments in CNT was the advent of the Johnson−Mehl−Avrami−Erofe’ev−Kolgomorov (JMAEK or simply ″Avrami″) kinetic model,33−39 developed in the late 1930s/early 1940s by the named crystal physicists, to describe the conversion mechanism termed “nucleation-and-growth” whereby the nucleation of spherical particles occurs randomly and isotropically throughout the medium simultaneously with the fixed-rate growth of the resulting nuclei up to the point of their impingement. That popular solid-state kinetic model (recently also appearing in the NP literature6,40), useful for describing the asymmetric sigmoid (S-shaped) conversion transients typically associated with phase transformations that are rate-limited by nuclei, lends support to the necessary connection between nucleation and NP growth that is inferable from CNT. Additionally, the more recent derivation of the generalized form of the JMAEK model by Bednarek et al.,41 using the concept of fractal conversion times, supports the idea that nucleation is a dispersive kinetic process. Experimental evidence of the greater multiplicity of energy states present in nanocrystal suspensions, relative to their dissolved monomer counterparts, can be found in the recent report of a 6 orders of magnitude enhancement in the variation of triplet excited-state lifetimes observed in substituted benzophenones in the nanocrystalline phase.42 Other literature supporting the idea that nucleation is a dispersive kinetic process can be found in ref 22. In comparison, deterministic kinetic models for crystal growth, which usually operate on the micrometer scale, typically assume a fixed rate constantbut none of them describe sigmoidal conversion versus time (x−t) transients.39 The biggest drawback of the generalized JMAEK model is its use of an empirical, unitless parameter28 to relate the fractal dimension (in comparison, the two fit parameters in the author’s dispersive kinetic model both have units and physically relevant descriptions,21 as will be discussed below). Another concern with that model is that it only relates “deceleratory” sigmoidal transients, in which the activation energy distribution is skewed toward lower energiesfrom the author’s recent work, it is clear that nucleation is an “acceleratory” process in which the activation energy distribution is oppositely skewed,28 as per the original derivation of his dispersive kinetic model that was based on the assumption of a Maxwell−Boltzmann (M-B)like distribution of activation energies.43 Those points made, the JMAEK model has remained a cornerstone in solidstate kinetics for many decades, and, as such, it is clearly more useful/meaningful in those applications than solution-phase reaction models that do not relate phase transformation (see the Supporting Information). Dispersive kinetics reflects conversion rates that are influenced by system relaxation. In such cases, one must employ a distribution of activation energies to allow the system dynamics to be captured in the conversion rate, in order to most simply and directly (i.e., without using numerical simulation) describe the overall kinetics of the process. Contrastingly, in cases where

2 x = e−[α / t ][exp(βt ) − 1]

(1)

where x represents the fraction of starting material remaining in the system at time, t (thus, the product fraction is related by p = 1 − x). The two fit parameters, α and β, have units of s and s−2, respectively; they have been assigned the following physical interpretations that support classical kinetic theories in the t-independent limit21 of small β: 0 ⧧ α = An − 1e2e−Ea / RT = An − 1e−ΔH / RT

β=

(2)

ΔS ⧧ Rt 2

(3)

where A is an Arrhenius-like frequency factor with the entropic factor explicitly removed from it, Ea0 is the Arrhenius-like (tindependent) portion of the activation energy potential, ΔH⧧ is the corresponding activation enthalpy, and ΔS⧧ is the tdependent activation entropy (such that ΔG⧧ = ΔH⧧ − TΔS⧧, whereby ΔG⧧ is also t-dependent due to the contribution from ΔS⧧), R is the universal gas constant, and T is the absolute temperature. In eq 2, n is a constant that relates the dimensionality of the system; in the derivation of eq 1, it was assumed that n = 0 because the free monomers leading to nuclei/NP formation are typically without dimension on the nanometer scale (as such, one might expect that the preexponential term has a much smaller value than would normally be expected for the collision frequency in chemical reactions, consistent with the widely accepted notion that nucleation is a low probability event). For acceleratory x−t sigmoidal transients produced via nucleation rate-limited kinetics, in which the rate of conversion is faster postinflection point (and beyond the induction period that is frequently observed at the outset of the process) than before it, β is inherently positive (in eq 3). It is responsible for producing the experimentally observable rate acceleration in the conversion transient, via lowering ΔG⧧ (the opposite is true for deceleratory conversions, as will be discussed in the next section). From eq 2 and eq 3, it is possible to deduce the following result for the t-dependent activation energy, Ea, in the case of nucleation: Ea = Ea0 − RT βt 2

(4)

The quadratic t-dependence predicted by eq 4 is very different from the logarithmic one predicted by the JMAEK equation,21,28 which is discussed more later. It was obtained 4844


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using the following functional form of the t-dependent rate coefficient (which is an approximation to the function that was integrated in order to obtain the closed-form model, eq 1):21

On the other hand, the t-dependent evolution of the mean NP size is related by the following equation:22 ⟨r ⟩ =

2 k = αeβt

(5)

Equation 4 and eq 1 can be used to obtain the following expression for the cumulative distribution function, cdf, of activation energies, f(Ea): ⎧ ⎪ βRT f (Ea) = exp⎨−α 0 ⎪ E a − Ea ⎩

⎤⎫ ⎡ ⎛ 0 ⎞ ⎪ ⎢exp⎜ Ea − Ea ⎟ − 1⎥⎬ ⎜ ⎟ ⎥⎦⎪ ⎢⎣ ⎝ RT ⎠ ⎭

(6)

⎫ ⎧ −U 2 ⎪Ye Q 2U[Y − 1] ⎪ −αQU[Y − 1] ⎬e g (Ea) = (αQ )⎨ − ; 2β ⎪ ⎪ U ⎭ ⎩ kBT ;

Q=

β Ea0 − Ea

;

2 Y = eβ /(QU )

(7)

where kB, the Boltzmann constant (kB ≡ R/N; N ≡ Avogadro’s number), is used in place of R from here on to relate atomic/ molecular quantities rather than molar ones. The distributions predicted by eq 7 are visually similar22 to the M-B distribution on which eq 1 was originally based, thus providing qualitative support for its derivation. Using the Kelvin equation, the author has previously shown22 that g(Ea) in eq 7 can be converted to a PSD, g(r), which is related by the following equation: ⎫ ⎧ −U 2 ⎪Ve F 2U[V − 1] ⎪ −αFU[V − 1] ⎬e g ( r ) = (α F )⎨ + ; 2β ⎪ ⎪ U ⎭ ⎩ 2 V = e−β /(FU )

βr ; 2γvm

F=

(8)

where r is the NP/nucleus radius, γ is the surface tension, and vm is the monomer volume. PSDs generated via eq 8 simulate NP distributions that are formed via a nucleation; they exhibit a distinct skew toward larger particle radii.22 That feature allows one to use the mechanism of OR, which gives rise to oppositely skewed PSDs (and which usually operates over longer time scales), to explain the phenomenon of PSD focusing.22 The width of the PSD can be related via the variance of the distribution, σr2, which is defined as σr 2 = r 2 − r

2 x = e[αt ][exp(−βt ) − 1]

∫0

r

2

(r − r )2 g (r ) dr

(9)

Ea = Ea0 + kBT βt 2

k = αe−βt

(10)

∫0

r

rg (r ) dr

2

(14) (15)

It is highlighted here that, in the derivation21 of eq 13, it was explicitly specified a priori that n = 2 (see eq 2). That dimensionality is thought to reflect the fact that denucleation is a surface-driven process46 and it results in different units (s−1) for α in eq 13 than in eq 1.

where the mean (expected) value of r, ⟨r⟩, is obtained by solving the integral ⟨r ⟩ =

(13)

In eq 13, the rate parameters, α and β, are defined as before, noting that β is explicitly shown to be negative in the model since it reflects the deceleratory nature of the empirical data. The rate deceleration, postinduction period, is due to a loss in the activation entropy as the conversion proceeds, which causes the activation energy barrier to increase and ultimately to slow down the conversion rate as a function of t. Using explicitly negative β values in eq 2 and eq 3 yields the following results:

From the pdf of particle radii defined by g(r) in eq 8, the variance is calculable as follows: σr 2 =

(12)

From eq 12 it is clear that the NP radius increases sharply as a function of t during non-steady-state nucleation, as the system supersaturation is relieved; hence, nucleation rate-limited processes described by eq 1 are appropriately termed “nucleation-and-growth” because they inherently capture the early t growth of the NPs in addition to the initial critical cluster formation. Such evolution in ⟨r⟩ is consistent with trends observed in the literature at early t, particularly in NP syntheses relying on burst nucleation; e.g. see ref 44. While the numerical simulations of Rempel et al.11 generally support that type of particle size evolution at early t (over the first three stages of NP formation, starting from monomer generation in solution), the PSD t-evolution was intentionally not treated in their work to simplify the modeling; dispersive kinetic models, such as eq 1, circumvent that issue by defining stationary PSDs, e.g., equation 8, akin to the PSDs associated with OR (discussed later). It is highlighted here that the simulation work of van Embden et al.12 also generally yielded a rapid increase in ⟨r⟩ during early t. Shevchenko et al.45 recently reported that CoPt3 NPs formed via hot synthesis “nucleate and grow up to their final size at an early stage of the synthesis”. That observation supports the notion that it might be possible, in some applications, that nucleation is the only mechanism that needs consideration in treating the monomer-to-NP conversion kinetics (eq 1), the NP size evolution rate (eq 12), or the NP PSD shape (via eq 8) assuming the precursor reduction step is rapid. 1.2. Denucleation. Denucleation is opposite to nucleation, whereby the formation of critical nuclei favors their dissociation rather than growth. As will be shown below, unlike in nucleation rate-limited conversions, during denucleation events the critical nucleus size decreases as the conversion proceeds (as the system moves toward solution saturation). To describe the deceleratory, sigmoidal x−t transients of denucleation rate-limited processes, one can use the following dispersive kinetic model that is complementary to eq 1:

The probability distribution function, pdf, of activation energies, g(Ea), is simply the differential of eq 6 with respect to Ea that can be written as follows:

U=

2γvm 0 Ea − kBT βt 2

(11) 4845


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The cdf of activation energies, f(Ea), is given by the following relationships: f (Ea) = e αQ ′ (Y ′− 1)/ U ; Q′ =

Ea0 − Ea ; β

U=

mechanisms in the sense that they describe phase transformations, the process of Ostwald ripening (OR),47−52 also known as particle coarsening, simply refers to the growth of larger, more thermodynamically stable particles at the expense of smaller (less stable) ones, as follows directly from the Kelvin equation. While nucleation and denucleation typically occur at the outset of NP formation (often occurring “on a relatively short time scale”5), OR typically operates over longer time scales.13,22 However, as OR does not represent a phase transformation mechanism, it does not make sense to define conversion-type models (akin to eq 1 and eq 13, presented earlier) for particle coarsening. Lifshitz−Slyozov−Wagner (LSW) theory49,50 was developed to predict the stationary/steady-state PSD shapes relevant to OR (analogous to the t-independent PSDs defined for nucleation and denucleation; see eq 8 and eq 18, respectively), as they pertain to both the diffusion rate-limited and reaction rate-limited cases of OR. In the diffusion rate-limited case, it is known from LSW theory that

kBT ;

2 2 Y ′ = e−β(Q ′) / U

(16)

The pdf of the activation energies, g(Ea), is related by ⎛ α g (Ea) = ⎜⎜ ⎝U β

⎞⎧ β Q ′Y ′ Y ′ − 1 ⎫ α(Q ′ / U )[Y ′− 1] ⎟⎟⎨ ⎬e − 2 β Q′⎭ ⎠⎩ U 2 ⎪

⎪

⎪

⎪

(17)

Plots of eq 17 for different values of the relevant physical parameters contained in the equation show that the distribution is skewed in the opposite manner22 to g(Ea) described by eq 7, giving rise to a kind of “inverted M-B-like distribution”.43 With use of the Kelvin equation, g(Ea) in eq 17 can be converted22 to the following PSD: ⎛ α g (r ) = ⎜⎜ ⎝U β F′ =

⎞⎧ β F ′V ′ V ′ − 1 ⎫ α(F ′ / U )[V ′− 1] ⎟⎟⎨ ⎬e − ; 2 β F′ ⎭ ⎠⎩ U 2 ⎪

⎪

⎪

⎪

2γvm ; β(r0 − r )

2 2 V ′ = eβF ′ / U

2γvm 0 Ea + kBT βt 2

g (u) = 0

otherwise (21)

(18)

where u ≡ r/⟨r⟩ and ⟨r⟩ is the mean NP radius. Similarly, in the case of the reaction rate-limited OR, g (u) =

273u 5

(2 − u)

e(−3u /(2 − u)) if 0 < u < 2

g (u) = 0

otherwise

(22)

Plots of eq 21 and eq 22 can be found elsewhere13 (additionally, the t-evolution of ⟨r⟩ and of the PSD are shown in other works12,13). In general, as per the denucleation case,22 the NP PSDs produced by coarsening are skewed toward smaller particle sizes; e.g., see ref 53. 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the theory indicates that mass flow in a biphasic system occurs from regions of high curvature to areas of lower curvature.54 However, it should be noted that recent works considering scenarios other than the limiting case treated by LSW theory (providing an asymptotic, analytic solution to the relevant physical equations in the theory) have yielded alternate PSD shapes52 that might make them difficult to distinguish from nucleation processes (e.g., if one does not simultaneously consider the time scales of the events). The simplest way to differentiate OR from nucleation/ denucleation might be to model the functionality of the tevolution of the mean NP radius size. From LSW theory it is known that, in the case of diffusion-controlled OR, the cube of ⟨r⟩, 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 ⟨r⟩ in reaction-controlled OR cases. The two relationships can be written mathematically as

(19)

From eq 19, it is clear that the NP radius decreases sharply as a function of t during non-steady-state denucleation (dissolution) processes, presumably as the system approaches monomer saturation. For both nucleation and denucleation, the evolution of ⟨r⟩ can be directly linked to the solution supersaturation level, whereby S* ≡ c/c∞ (c is the dissolved monomer concentration at time t, c∞ is the bulk solubility, and S*0 is the initial supersaturation) at every point in time, using the following relationships: ⎡ ⎛ S * ⎞⎤ 2 2γvm ⎢ln⎜ 0 ⎟⎥ ∝ = Ea0 ± kBT βt 2 ⎢⎣ ⎝ S* ⎠⎥⎦ r − r0

if 0 < u < 1.5

(u + 3)7/3 ((3/2) − u)11/3

where r0 is the initial NP/nucleus radius size at t = 0 (the start of the conversion). The variance of the distribution is defined as per eqs 9−11 (note that the width and shape of the PSD defined by eq 18 can also be important in understanding focusing behavior of the type that can result from simultaneous nucleation and denucleation22). The t-dependent evolution of the mean NP size is given by the following equation:22 ⟨r ⟩−⟨r0⟩ =

34 e/25/3)u2e[−1/(1 − (2u /3))] ( g (u) =

(20)

where the sign on the right hand part of the equation is dependent on the type of mechanism. Equation 20 might be useful in modeling cases where the solution concentration of monomers is monitored as a function of t (e.g., using online FTIR spectroscopy, for processes involving organic compounds as the monomers), as opposed to the particle size/PSD evolution or some conversion metric pertaining to the solid state (e.g., using online Raman spectroscopy, in the case of phase changes involving organic compounds). 1.3. Ostwald Ripening (Particle Coarsening). While nucleation and denucleation processes are true conversion

1/3 ⎡ 8γc∞vm 2D ⎤ 3 ⟨r ⟩ = ⎢⟨r0⟩ + t⎥ ⎢⎣ ⎥⎦ 9kBT 4846

(23)


Langmuir 1/2 ⎡ 64γc∞vm 2k ⎤ 2 t⎥ ⟨r ⟩ = ⎢⟨r0⟩ + ⎢⎣ ⎥⎦ 81kBT

Article

different materials, including polystyrene, silica, and gold colloids.59 In the case of DLCA, the cluster mass distribution is essentially independent of M up to the average mass, ⟨M⟩, beyond which it falls off exponentially. Solving the Smoluchowski rate equation,60 assuming that the NPs stick to each other immediately and permanently upon initial contact, the cluster mass distribution, f(M), corresponding to DLCA can be shown to be well-approximated using the following relationship:61

(24)

where c∞ is the monomer solubility, D is the diffusion coefficient (where D = kBT/6πηa, η is the solvent viscosity, and a is the monomer hydrodynamic radius), and k is the reaction rate constant. The temporal evolution of ⟨r⟩ described by eq 23 and eq 24 is clearly very different from that described by either eq 12 or eq 19. That distinguishing factor is made even more important by the fact that one cannot infer OR behavior by monitoring conversion kinetics, as mentioned earlier. From a practical perspective, the mean NP volume, ⟨V⟩, should not be plotted directly as a function of t, in place of ⟨r⟩3 (in eq 23), because ⟨V⟩ ∝ ⟨r3⟩ and not ⟨r⟩3. While the important role of OR in current NP preparations has been discussed by numerous authors,9,13,15,16,55,56 in other works the empirical data (e.g., for the t-evolution of ⟨r⟩ and for the fractional conversion)5,6 might also support OR over longer time scales. Over long t periods, OR leads to defocusing16 and the observation of asymmetric particle distributions;13 however, the more rapid increase in particle size as t → ∞, predicted by some recent numerical simulations,11,12 does not appear to be consistent with an OR-type mechanism. 1.4. NP Aggregation. The aggregation (flocculation) of NPs has been reported widely in the recent literature.3,6,57,58 Aggregation might be an important process, e.g., in the formation of nanocomposites, because it allows one to achieve that next level in the material hierarchy. Like OR, aggregation is not a conversion mechanism, thus requiring one to monitor the t-evolution of the average mass or size of the aggregate instead of the conversion kinetics. An exception might be the case of protein aggregation (discussed later), whereby the soluble protein molecules are converted to insoluble aggregates by just that mechanism, precluding nucleation-and-growth that is more often associated with cases where the monomers are atoms or small molecules. Frequently, the average cluster mass, ⟨M⟩, is used in lieu of ⟨r⟩ in aggregation studies because aggregates often exhibit fractal geometries that are not spherical. M scales with the aggregate hydrodynamic radius according to the equation:59

⎛ R g ⎞n M∝⎜ ⎟ ⎝ r ⎠

f (M ) =

M−1 N0 ⎛ 1 ⎞ 1 − ⎜ ⎟ 2⎝ M ⎠ M

(26)

where N0 is the total solid-state mass in the system. As per the author’s PSDs for nucleation and denucleation, as well as the PSDs defined by LSW theory for OR, eq 26 represents a steady-state mass distribution whose functionality is independent of time. The Smoluchowski equation also predicts a linear t-dependence of the average cluster mass for DLCA, according to the following equation:61 M = kat + 1

(27)

where ka is the aggregation rate constant multiplied by the starting colloid concentration (ka ≡ 8kBTN0/3ηV; the ratio, N0/V, represents the initial NP concentration, and η is the viscosity of the medium). Empirically linking the average cluster radius of gyration, ⟨Rg⟩, to the mean hydrodynamic radius, ⟨Rh⟩, and using the above equations, one obtains the following power-law relationship describing the t-evolution of ⟨Rh⟩ in DLCA cases:61 R h = 1.44r(kat )1/1.8

(28)

Thus, the t-dependence of the cluster radius evolution in DLCA is closest in functional form to eq 24, describing the case of reaction-controlled OR. However, as OR does not produce aggregates, the two mechanisms can likely be distinguished by empirical observation of the product solids. In the case of RLCA, the cluster mass distribution has a functional form that, for a wide variety of kernels used in the solution of the Smoluchowski equation, is given by a power law with an exponential cutoff:62 f (M ) =

(25)

N0Mc τ− 2M −τe−M / Mc Γ(2 − τ)

(29)

where Γ(v) is the gamma function, Mc is the cutoff mass, and τ ≈ 1.5. It has been shown62 that a universal relationship for the growth of the mean cluster hydrodynamic radius is a simple exponential in time:

where Rg is the radius of gyration of the cluster, n is its fractal dimension, and r is the radius of the individual NPs, assumed to be present in a monodisperse suspension at the outset of the aggregation event. In diffusion-limited colloid aggregation (DLCA), n ≈ 1.8, and in the case of reaction limited colloid aggregation (RLCA), n ≈ 2.1. DLCA is akin to spinodal decomposition in that there is essentially no activation energy barrier that needs to be overcome to allow two colloids to stick to each other upon collision (unlike RLCA, in which there is such a barrier); that mechanism typically dominates at higher ionic strengths where the electrostatic repulsion between colloids caused by the double layer is weakest. DLCA typically produces better-organized aggregates than RLCA and, expectedly, does so at a faster rate. The two processes are understood quite well, and both are universal in that the relationships presented below have been shown to generally hold for very

R h ∼ekat

(30)

RLCA typically dominates under low ionic strength conditions and it often leads to the formation of more randomized aggregate structures. In any event, tracking the evolution of the cluster radius size as a function of t allows one to distinguish between RLCA and DLCA for a given NP aggregation process simply by comparing the curve-fit qualities of both eq 28 and eq 30 to empirical ⟨Rh⟩ vs t data. However, it might be harder to distinguish particle growth via RLCA from nucleation (eq 12), at early t, provided limited data. Case-inpoint, in chiral resolutions involving conglomerate formation, 4847


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the rate-limiting step might be secondary nucleation63 or “Viedma ripening”.64 The latter mechanism is akin to RLCA in that it involves the attachment of small clusters. Because of the similarity in the particle growth kinetics between nucleation and RLCA (formerly considered nucleation via units other than monomers22), one can draw a parallel between the two processes.

In RLCA, Ea,2 can be defined with the aid of the theory of Derjaguin, Landau, Verwey, and Overbeek.67,68 DVLO theory accounts for both the van der Waals attraction and electrostatic repulsion that is due to the colloid double layer. For the simplest of systems consisting of two spherical NPs one can define the work of interaction between them, W, using the Derjaguin approximation:

■

RESULTS AND DISCUSSION 2.1. Physicochemical Interpretation of the Activation Energy of Aggregation. For the process of DLCA, ka was defined in the previous section. While DLCA is often considered a barrierless process, a parallel being drawn to spinodal decomposition (as described by Cahn−Hilliard theory),65 by considering the T-dependence of the NP diffusion constant, D, which generally exhibits classical Arrhenius behavior [D = D0 exp(−Ea,1/kBT)] in the long t limit, and the relationship that exists between the viscosity of the medium, η, and D (i.e., η = DmN0/V, where m is the NP mass), one can provide a more detailed description of ka, as follows: ka =

8kBT Ea,1/ kBT e 3mD0

W ∼ Ea,2 =

8kBT ((Ea,1− Ea,2)/ kBT ) e 3mD0

(33)

where Λ is the Hamaker constant (Λ ≡ π2C′ρ1ρ2, where the ρ values reflect the number density of monomers in each particle and C′ is the coefficient in the van der Waals particle−particle pair interaction potential that decays as d−6; Λ has units of energy), d is the separation distance between the two colloids, and r1 and r2 represent the two NP radii. From eq 33 one can infer that the interaction force is proportional to the inverse square of d, making it possible to draw a parallel between that force, between two NPs, and the gravitational force between two celestial bodies. Following the derivation of Kepler’s first law of planetary motion, but using a DVLO-type interaction force (based on eq 33) in place of the gravitational force (and neglecting NP drag forces,69 for simplicity), one can envision that it might be possible to derive an equation defining a stable orbit. Presumably, under ideal conditions, two similarly sized NPs might begin a mutual (circular) orbit, with the center of mass serving as the focus; on the other hand, for a NP orbiting a large aggregate, the orbit might be more elliptical, with the aggregate serving as one of the two foci. Using this interpretation, provided a NP is on a collision course with an aggregate, one can imagine that their collision can cause (further) aggregation if the NP has sufficient energy to overcome Ea,2 (as per RLCA; see eq 32) but not so much energy as to subsequently detach. If its kinetic energy is too large, the escape velocity will be achieved (producing a stable orbit) or exceeded (yielding an unstable orbit, resulting in NP deflection) and, consequently, no aggregation will take place. The escape velocity, ve, can be estimated using the following equation:

(31)

where Ea,1 is the activation energy barrier for colloid diffusion and D0 is the colloid diffusion constant at the reference temperature, T0. As the system temperature is raised, ka diminishes. That reduction in the aggregation rate constant reflects the situation whereby the colloids/NPs having too much kinetic energy to stick together and form an aggregate, via DLCA. In the RLCA scenario, there is an interaction potential between the NPs that must be overcome in order to allow the aggregate to form. Using Ea,2 to represent the energy barrier that needs to be overcome in order to observe aggregation via RLCA, one obtains the following relationship for ka: ka =

−Λ ⎛ r1r2 ⎞ ⎟ ⎜ 6d ⎝ r1 + r2 ⎠

(32)

According to eq 32, ka increases rapidly as a function of T, provided that Ea,2 > Ea,1 (as expected for RLCA). In the special case where Ea,2 = Ea,1, ka increases linearly with T and when Ea,2 < Ea,1, ka reflects the DLCA-type behavior described by eq 31. In plots of ka vs T (not shown), the fact that the specific rate of aggregation can either increase or decrease with T (depending on whether RLCA or DLCA is the dominant mechanism) gives rise to a profile having a peaked appearance at intermediate temperatures. Such behavior is akin to nucleation rate behavior in the limit of spinodal decomposition,30 in that nucleation is generally faster at lower temperatures (where greater system supersaturation provides a stronger driving force for nucleation), whereas spinodal decomposition is facilitated by higher T. It has been noted that “samples often show behavior intermediate between RLCA and DLCA”,65 and, furthermore, it has been proposed66 that there might be an additional universal behavior in that colloidal aggregations, at moderate ionic strength, always start as RLCA and eventually transition over to DLCA. Growth via DLCA is typically faster, while RLCA is usually sluggish; the transition from RLCA to DLCA observed at intermediate salt concentrations might occur due to the fact that the initially formed clusters (via RLCA) improve the cluster sticking probability, which ultimately leads to DLCA.

ve =

Λ ⎛ r1r2 ⎞ ⎟ ⎜ 3md ⎝ r1 + r2 ⎠

(34)

2.2. Time-Evolution of the Mean Radius Size in Bismuth Nanocrystals. In a recent work,56 kinetic data were presented for the evolution of the mean NP volume, ⟨V⟩, as a function of t, obtained during the preparation of bismuth nanocrystals. In Figure 1 of this paper, a data set from the original work was digitized and then replotted approximately as ⟨r⟩ vs t (see section 1.3 of the Introduction; note also that the assumption, herein, of a spherical particle geometry does not invalidate the present approach since one can define an appropriate geometric factor, serving as a coefficient in the CNT expression for the activation energy, to correct for the actual NP geometry) and then fit to the relevant kinetic models presented herein. It was found that the transient might be adequately described by eq 12 at early t, reflecting a nucleationand-growth mechanism, and eq 23 over later t, reflective of diffusion-limited OR. However, one cannot entirely preclude an RLCA mechanism at early t (eq 30)70 nor a reaction-limited OR mechanism (eq 24) at later t, because the scatter in the data may not allow one to distinguish the mechanisms entirely on 4848


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the author has recently discussed that modeling NP kinetic trends should be done using the majority fraction (as determinant of the rate-limiting mechanism, at any given t) due to the possibility of overlapping/competing mechanisms.22 As a point of interest, the growth of Bi NPs appears to involve a longer nucleation period56 than typical Au NP preparations5 (and thus it is more akin to the traditional solidstate conversions studied previously by the author21). Also, there is no quasi-stationary period in the data presented in Figure 1, unlike the predictions of some numerical models presented in the literature (see Introduction). Perhaps, the quasi-stationary region applies mainly to cases involving burst nucleation, which very rapidly depletes the immediate monomer supply, leading to conditions whereby the solution reaction (reduction) step can also become rate-limiting (particularly when weaker reducing agents are used). In such cases, the monomer production and simultaneous monomer consumption (via nucleation) can potentially give rise to steady-state nucleation conditions and, consequently, produce monodisperse NP distributions simply by providing conditions that support the formation of a single-sized critical nucleus.54 2.3. Protein Aggregation Kinetics and the Connection to Population Survival, Tumor Growth, and Neurological Disease. In two recent papers, the role of protein aggregation in the aging of Caenorhabditis elegans (C. elegans)73 and in the development of malignant tumors74 was discussed. Previously, the formation of insoluble protein aggregates was discussed mainly in the context of neurodegenerative diseases like Alzheimer’s disease,75 Huntington’s disease,76 Parkinson’s disease,77 and other (prion-related) disorders.78 It is thought that by better understanding the mechanism of protein aggregation one might gain important insight into the processes of aging, the occurrence of cancer, and the development of neurological disease. In that regard, many workers have endeavored to understand the behavior over the past 50 years.79 As an aside, in terms of pharmaceutical development, the prevention of protein aggregation in treatments including insulin80,81 and the recombinant protein/biosimilar, erythropoietin,82 continue to receive attention because the aggregation causes inactivity. The connection between protein aggregation and lifespan, recently proposed by David et al.73 in C. elegans, might be an important development in advancing fundamental mechanistic understanding of aging. Their finding might also provide justification for use of the Gompertz equation83 in protein aggregation kinetic modeling applications, since the Gompertz function has long been used to describe population mortality and, more recently, malignant tumor formation.84,85 The hypothesis that will be tested in this section of the paper is: does the Gompertz function have a physicochemical basis in kinetic studies of protein aggregation? If such a link is indeed found, one might then ask the question: is lifespan and malignant tumor formation/growth (in at least some cases), like certain neurodegenerative diseases, rate-limited by a protein aggregation mechanism? The Gompertz model can be written as

Figure 1. Plot of the relative radius size evolution, over time, for the formation of bismuth nanocrystals in a kinetic trial conducted using a Na[N(SiMe3)2] concentration of 0.049 M. The data points were extracted from Figure 5 of ref 56, and the mean NP volumes were converted to approximate NP radii assuming a spherical geometry. The solid blue line is a regression fit of the data to eq 12 in the text (R2 = 0.737), providing qualitative support for a nucleation rate-limited mechanism at early t. The solid green line is a regression fit to eq 23 in the text (R2 = 0.973), providing support for an OR rate-limited mechanism at later t. Over the intermediate range, 35 min < t < 95 min, the evolution of r is thought to involve both nucleation-and-growth and OR mechanismsthat mechanistic transition window has been discussed previously to provide a viable route to PSD focusing.22 The broken lines represent crude extrapolations of the two models and are intended as visual guides for the reader; their intersection represents the approximate mean transition time between the two mechanisms.

the basis of curve-fittingthat is where qualitative information (e.g., high-resolution transmission electron microscope (HRTEM) micrographs) and a priori knowledge about the process (e.g., that nuclei or clusters first need to form before they can aggregate, that RLCA usually dominates over shorter time periods than DLCA, and that NP aggregation does not contribute to particle focusing) can help to differentiate the mechanisms. Kinetic data obtained at intermediate time points has contributions from both mechanisms (those data were intentionally not fit, for clarity of presentation and to avoid use of constructs, e.g. an “autocatalytic turn-on function”, to bridge the two time periods), thus providing a means of sharpening the particle size distribution, i.e., PSD focusing. Interestingly, the recent work by Wu et al.10 supports the idea of PSD focusing, in the case of Au19 clusters, that occurs via the combination of nucleation rate-limited and OR rate-limited processes, as theorized by this author.22 Other workers have similarly discussed a mechanistic crossover, e.g., between collision-coalescence (also called oriented attachment)/RLCA and OR70,71 or, alternatively (more along the lines of the mechanism proposed in this work), from reduction/nucleation/particle growth to diffusion-limited OR.72 The JMAEK model was intentionally not used here because Figure 1 does not relate a conversion transient. Additionally, as will be discussed in section 2.4, the JMAEK model predicts only the formation of smaller nuclei/NP sizes as a function of t, which is inconsistent with the data in Figure 1. Finally, a NP aggregation mechanism was not utilized in this work since it was estimated in the original paper that only 20−40% of the NPs observed via HRTEM after ∼10 min were polycrystalline56 (and, thus, potentially supportive of an aggregation mechanism);

x = e(α / β)[exp(−βt ) − 1]

(35)

where x traditionally represents the fractional population alive at a given t (hence the terminology “survival function” for that equation) and the empirical curve-fitting parameters, α and β (note: the latter is explicitly negative in both eq 35 and eq 36), both have units of inverse time. The complementary, “p-form” of eq 36 can be used in modeling tumor growth. Interestingly, 4849


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ultimately results in a lower system free energy. In fact, the same effect has recently been demonstrated in the molecularcrowding-enhanced aggregation of DNA-wrapped carbon nanotubes.94 While the excluded volume effect, e.g., see ref 95, is not treated by current RLCA theory, the work of Hobbie96 also suggested its importance in aggregation events. To account for it most simply, one can consider that the process is dispersive and, consequently, that Ka is not a constant but, rather, it is t-dependent. Introducing a power-law tdependence to Ka, where the power is a priori set at a value of 3, eq 38 becomes

eq 36 can also be found in descriptions of quantum dot (QD) electron-transfer kinetics86 that exhibit Poisson statistics87 (as per the work of Tachiya88,89). Furthermore, the model has appeared in the recent literature describing the sigmoidal population growth dynamics of carbon nanotubes.90 Equation 36 describes a first-order conversion mechanism (F1) with a t-dependent specific rate, k, undergoing an exponential decrease as a function of t, as per the equation: k = αe−βt

(36)

By substituting eq 36 into the integrated expression for a dispersive F1 mechanism, given by x = exp( −

∫0

t

k dt )

−5.1 I = e−Kat I0

(37)

The curve fit of eq 39 to the data set in Figure 2C is dramatically improved relative to that of eq 38, even with the C term explicitly removed from it. Analogous to eq 39, the unusually large Avrami exponent39 reported in a recent work describing Au NP growth kinetics,97 yielding values around 6, might similarly reflect the “aggregative growth” of larger NPs (in the 10−100 nm range) starting from three-dimensional (on the nanometer scale; recall that atoms/small molecules do not have dimensions on this scale) primary particles. A scaling dimensionality of 3 is also known for spinodal decomposition.66 In any event, the potential for dispersive kinetic behavior in certain aggregation processes is supported by the known restricted motion of colloids in aggregate structures (provided that their sticking together is not permanent) and, hence, the slow/complex evolution of aggregates to reach their equilibrium phases.65,66,95 Thus, certain cases of protein (and NP) aggregation might represent a class of dispersive kinetic processes whereby a true rate constant is only observed if the system relaxation is fast compared to the time scale of the aggregation rate. In presenting Figure 2, the author does not intend to suggest that better curve fits cannot possibly be obtained using other models,79,98 in certain cases, only that the curve fits of those data sets to the (one or) two fit-parameter-containing model, (eq 39 or) eq 38, seem to be quite good with regard to the scatter inherent in the experimental data; consequently, the values (of Ka) obtained have sufficiently low errors associated with them to be usable for prediction/additional data interpretation. More importantly, the models presented herein have a sound physicochemical basis in RLCA theory, and they are very simple, yet they correctly model the expected signal intensity changes for processes that have long been reported to be the result of protein aggregation, based on empirical observations. Specifically with regard to the applicability of nucleation-based models, it is noted here simply that even for a small protein such as α-synuclein (containing 140 residues), if the smallest critical nucleus were assumed to be the native “dynamic tetramer”99 (which is less prone to aggregation than the mutated/denatured protein), then its diameter would already be ∼10 nm, thus exceeding the typical size of most metallic (e.g., Au) and semiconductor (e.g., CdSe) NP preparations described in current literature, despite its containing only four monomer units. In DLCA cases, the equivalent expression to eq 38/eq 39 is

and solving eq 37, it is trivial to derive eq 35. While the Gompertz model is well-understood,84,91 the relationship in eq 36 has remained an empirical one to this day (to that end, in section 2.4 of this paper the model will be further scrutinized from a physicochemical standpoint). For current purposes, it is simply noted here that a t-dependent k is a property of all dispersive kinetic models, which stems from an underlying distribution of activation energies (see Introduction). The similarity between the functional forms of eq 36 and eq 30 is intriguing; especially since the Kelvin equation can be used to (approximately) relate the aggregate radius size evolution via RLCA to a t-dependent Ea, in turn, yielding a tdependent rate coefficient of the same functional form as eq 36. Consequently, there indeed might be a parallel between the Gompertz and RLCA models. In that regard, the existence of a possible connection between protein aggregation kinetics and the kinetics associated with population lifespan/tumor growth/ neurological disease has clearly not been disproven. Studies of protein aggregation kinetics typically monitor fluorescence or light scattering signal as a function of t. Starting from eq 25 and eq 30, assuming that n ≈ 2.1 and taking into account the fact that the static light scattering intensity, I, trends with the mean particle radius59 as a function of ⟨R⟩−n, it is possible to deduce that −2.1 I = Ce−Kat I0

(39)

(38)

where I0 is the starting signal intensity, C is an empirical constant that simply aids in certain curve-fitting applications (often unnecessary; see values close to unity in Figure 2), and Ka has been capitalized to highlight its modification from the previously defined rate constant, ka. Equation 38 might also find use in fluorescence studies; as support for that idea, in the case of Ag NPs the scattering properties of the plasmons are known to be dominant factors in obtaining the fluorescence signal enhancements observed at smaller particle sizes.92 Indeed, from Figure 2 it is clear that eq 38 fits real-world protein aggregation data quite well, regardless of whether light scattering or fluorescence measurements were used to monitor the kinetics. The clear exception is scenario C of Figure 2. With regard to case C in Figure 2, it has long been known that the use of molecular crowding agents such as polyethylene glycol (PEG) can help induce the precipitation of proteins,93 via an overlapping effect in the excluded volumes of the protein molecules that results in a greater free volume available to the PEG molecules (which is accompanied by an increase in system entropy, providing thermodynamic driving force) that

I = K at n I0 4850

(40) dx.doi.org/10.1021/la205034u | Langmuir 2012, 28, 4842−4857

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Figure 2. Representative plots of neurological protein aggregation kinetics. The data were obtained using either fluorescence or light scattering detection. The violet lines represent regression fits using eq 38 for RLCA. The red curve represents a fit using eq 39, which is thought to reflect dispersive RLCA behavior. The plots are as follows: (A) data set from ref 117 for amyloid β peptide aggregation; R2 = 0.957, C = 1.43 ± 0.17, and Ka = (1.05 ± 0.19) × 104 h2.1. (B) Data from ref 118 for the aggregation of amyloid β peptide; R2 = 0.962, C = 1.014 ± 0.026, and Ka = (1.29 ± 0.21) × 103 min2.1. (C) Data from ref 119 for α-synuclein aggregation in the presence of the macromolecular crowding agent polyethylene glycol; R2 = 0.951, C = 1.123 ± 0.017, and Ka = (1.15 ± 0.08) × 103 h2.1 (violet curve), and R2 = 0.995, Ka = (−2.71 ± 0.09) × 107 h5.1 (red curve). (D) Data from ref 120 for α-synuclein aggregation; R2 = 0.952, C = 1.10 ± 0.08, and Ka = (5.5 ± 1.4) × 102 h2.1. (E) Data from ref 121 for polyglutamine aggregation; R2 = 0.983, C = 1.42 ± 0.10, and Ka = (2.8 ± 0.3) × 103 h2.1. (F) Data from ref 122 for D67H human lysozyme aggregation; R2 = 0.993, C = 0.958 ± 0.004, and Ka = 1.466 ± 0.025 h2.1. Except for case C, the limited precision of certain curve fits can be attributed largely to the scatter in the experimental data (error bars not shown for clarity). In scenario C, the enhancement in aggregation rate caused by molecular crowding is not adequately treated by current RLCA theory; for that reason eq 39 was used (see text for details).

where the exponent, n, is left general. Interestingly, eq 40 was unable to precisely model any of the data sets presented in Figure 2, suggesting that DLCA is not as important a mechanism in neurological protein aggregation

as RLCA (or dispersive RLCA). In any event, the author does not think that invoking nucleation and/or other mechanisms is warranted to explain the kinetics of those processes. 4851


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2.4. Distribution of Activation Energies Underpinning a Recently Reported Synthesis of Copper Indium Sulfide Nanocrystals. As mentioned in the Introduction, kinetic and thermodynamic understanding of NP processes is essential for developing/optimizing robust synthetic approaches. In a recent paper,100 the efficient synthesis of copper indium sulfide (CIS) nanocrystals (NCs) was described. In this section, the kinetics of that process is discussed in light of various dispersive treatments, in order to gain insight into the key (rate-limiting) mechanism underpinning the synthesis. In Figure 1b of ref 100 the kinetics of the CIS NC formation was monitored via the collection of emission spectra over time. Using the photoluminescence (PL) peak energy, E, plotted as a function of t (Figure 3) and by considering that trend to serve

From Figure 4 it can be seen that eq 43 yields activation energy distributions that resemble some of those presented in the previous dispersive kinetics literature.19−21,28 However, the Gompertzian g(Ea) plots tail in a manner that is closer to denucleation than nucleation cases (but consistent with the notion that “biological growth, as empirically known, always eventually slows down”84). As an aside, the different distribution areas that can be inferred from Figure 4, also predicted for the author’s

Figure 4. Theoretical plots of g(Ea) vs Ea obtained using eq 43, for various values of the relevant physical parameters: (red) α = 0.01 s−1, β = 0.002 s−1, T = 298 K, Ea0 = 85 kJ/mol; (black) α = 0.011 s−1, β = 0.002 s−1, T = 298 K, Ea0 = 85 kJ/mol; (blue) α = 0.01 s−1, β = 0.003 s−1, T = 298 K, Ea0 = 85 kJ/mol; (green) α = 0.01 s−1, β = 0.002 s−1, T = 298 K, Ea0 = 80 kJ/mol; (violet) α = 0.01 s−1, β = 0.002 s−1, T = 498 K, Ea0 = 85 kJ/mol.

Figure 3. Scatter plot of E vs t (●) using the data in Figure 1b of ref 100. The linear regression fit has R2 = 0.976, intercept of 183.1 ± 1.5 kJ/mol, and slope of −0.45 ± 0.04 kJ mol−1 min−1.

models,22,28 likely reflect different probabilities of formation for the differently sized critical nuclei. Analogously, the work of van Embden et al.12 (see Figure 9E in that reference) similarly showed different PSD areas at different supersaturations, suggesting that more NPs are likely to form when there is a greater thermodynamic driving force for nucleation, consistent with expectation. Instead of directly using the PL peak energies as per Figure 3 of this work, the ΔEa values plotted in Figure 5 were estimated using the mean particle radii plotted in Figure 1a of ref 100. Those radii were converted to Ea values using the Kelvin relationship, assuming that γ = 1.7 J/m2, as per the value101 for Cu, and νm = 0.02 nm3, assuming an average monomer radius of 0.17 nm (which is the effective radius102 of S2− as well as the average van der Waals radius of both In and Cu). The data points in Figure 5 are best fit by a simple logarithmic function that is more consistent with the generalized form of the JMAEK equation than the Gompertz model. Bednarek et al.41 derived the following relationship for the former model:

as a reasonable surrogate for the t-dependent Ea of the conversion, then the following relationship might be used to fit the data: Ea = Ea0 − kBT βt

(41)

Because eq 41 follows directly from eq 36, one might propose to investigate the utility of the Gompertz model in nonbiological applications such as the present one. However, in doing so, it might be worthwhile to first determine if eq 35 can be used to predict a realistic Ea distribution, which has not been examined before in the previous literature, to the author’s knowledge. Using eq 35 and eq 41, one can obtain the following cdf of activation energies: ⎧ ⎫ ⎡ ⎛ 0 ⎤⎪ ⎪ E − Ea ⎞ ⎟⎟ − 1⎥⎬ f (Ea) = exp⎨( −α /β)⎢exp⎜⎜ a ⎢⎣ ⎝ kBT ⎠ ⎥⎦⎪ ⎪ ⎭ ⎩

(42)

⎛ t ⎞ Ea = Ea0 + (1 − n)kBT ln⎜ ⎟ ⎝ τ0 ⎠

Differentiating f(E a ) with respect to E a yields the corresponding pdf, given by g (Ea) =

⎪ Ea0 − Ea ⎤⎫ ⎥⎬ kBT ⎥⎦⎪ ⎭

(44)

In eq 44, τ0 is a time constant (akin to a reciprocal rate constant) and n is an empirical parameter relating the system dimensionality. When coupled with a first-order conversion mechanism, consistent with the derivations of eq 1, eq 13, and eq 35, the generalized JMAEK model is obtained. The activation energy distributions, g(Ea), corresponding to that model can be found elsewhere;19 they are visually similar to the curves

⎧ α ⎪ ⎡⎢ −α ((Ea0 − Ea)/ kBT ) ⎨exp (e − 1) βkBT ⎪ ⎩ ⎢⎣ β +

0

Roles of Nucleation, Denucleation, Coarsening, and Aggregation

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