ARTICLE pubs.acs.org/JPCA
Activation Energy Distributions Predicted by Dispersive Kinetic Models for Nucleation and Denucleation: Anomalous Diffusion Resulting from Quantization Peter J. Skrdla 640 Maple Street, Westfield, New Jersey 07090, United States ABSTRACT: The activation energy distributions underpinning the two complementary dispersive kinetic models described by the author in a recent work (Skrdla, P. J. J. Phys. Chem. A 2009, 113, 9329) are derived and investigated. In the case of nucleation rate-limited conversions, which exhibit “acceleratory” sigmoidal transients (a kind of S-shaped stretched exponential conversion profile), an activation energy distribution visually similar to the MaxwellBoltzmann (MB) distribution is recovered, consistent with the original derivation of that model. In the case of predominantly “deceleratory” conversions, the activation energy distribution is skewed from normal in the opposite direction. While the “MB-like” activation energy distribution supports the empirical observation of a rate enhancement as a function of the conversion time in nucleation rate-limited processes, the complementary distribution, with its pronounced low-energy tail, reflects a slow-down in the specific rate as the conversion progresses, consistent with experimentally observed denucleation rate-limited conversions. Activation energy distributions were also plotted for real-world data (Qu, H.; Louhi-Kultanen, M.; Kallas, J. Cryst. Growth Des. 2007, 7, 724), depicting the impact of various additives on the nucleation rate-limited kinetics of the solventmediated phase transformation of the crystalline drug carbamazepine. Last, by coupling the author’s dispersive kinetic description of the time-dependent activation energy for nucleation to the classical description of the critical nucleus energy provided by the Kelvin equation, an accelerated hopping mechanism for the diffusion of monomers to the growing embryo surface was observed. That hopping mechanism was rationalized by modifying the EinsteinSmoluchowski (ES) equation to allow it to describe the “supraBrownian” molecular motion thought to lie at the heart of nucleation kinetics.
’ INTRODUCTION Nucleation is a mechanism that can be observed all throughout nature. Denucleation, the less well-known but complementary process, is likely just as important. Efforts aimed at accurately describing and predicting nucleation rates, which began in earnest in the last century, continue to the present day, in large part driven by the recent revival of interest in colloids (i.e., nanoparticles, nanocrystals). Despite the numerous scientific and technological advancements made over the years, much of the current practical (deterministic) understanding of nucleation kinetics does not extend far beyond classical nucleation theory (CNT) or use of the JohnsonMehlAvramiErofe’evKolmogorov (JMAEK or Avrami) equation. In this work, the over-70-yearold JMAEK model is revisited in the context of dispersive kinetics theory. A parallel is drawn to the analogous theory recently proposed by the author in the development of two other dispersive kinetic models, one for treating nucleation rate-limited conversions and the other for denucleation ratelimited ones, both of which have previously been shown to be useful in both modeling and understanding the complex kinetic profiles of real-world processes. The foundational assumption used in the derivation of those models, the existence of a r 2011 American Chemical Society
MaxwellBoltzmann-like (MB-like or inverted MB-like in the denucleation case) distribution of activation energies is probed in an effort to determine whether or not the derivation of those models is self-consistent with the proposed theory. In other words, the question posed is, What do the activation energy distributions predicted by each model really look like? (Are distributions indeed mathematically recoverable from them?) Last, the author’s dispersive kinetic model for nucleation rate-limited conversions is coupled with a key CNT-based equation describing the size-dependent energy of the critical nucleus in an effort to gain fundamental insights into the molecular-level dynamic behavior occurring at the nucleus surface. 1. Background. Dispersive kinetics,1 sometimes referred to as distributed kinetics or stochastic kinetics, is underpinned by the concept of a distribution of activation energies. The activation energy distribution (as opposed to the single activation energy that is typically inferred via inspection of the Arrhenius/Eyring Received: December 10, 2010 Revised: May 12, 2011 Published: May 19, 2011 6413
dx.doi.org/10.1021/jp111767c | J. Phys. Chem. A 2011, 115, 6413–6425
The Journal of Physical Chemistry A
ARTICLE
equation) is simply a manifestation of system heterogeneity that can cause, for example, a reactant species to become less reactive as a given conversion proceeds.2 Such a change in the apparent reactivity can come about as a result of local structural relaxation that occurs in parallel with the overall conversion; these continuous environmental renewals (i.e., system dynamics) can impact the kinetics if they occur on a time scale that is similar to, or slower than, the conversion under investigation. Note, however, that in the event that the system renewals occur significantly faster than the conversion, the dispersive kinetics paradigm simply reverts back to the classical kinetics one, whereby the activation energy distribution collapses into a Dirac delta function (as such, the development of rational dispersive kinetic treatments should account for that behavior). Dispersive effects have been reported in the spectroscopy of single molecules3,4 (even of those in highly ordered environments, such as on crystal surfaces studied under UHV conditions5) and, similarly, in the nonexponential (i.e., non-MaxwellDebye relaxation) blinking kinetics of quantum dots.6 Relaxation times in polymeric systems, crystal thermolyses, diffusion-limited reactions, protein folding, certain crystallizations, and vitrification can also exhibit dispersive kinetics.1,7 Plonka7 has stated that “while the KohlrauschWilliamsWatts (KWW; vide infra) stretched exponential and Curievon Schweidler (CvS) power law have... been used to characterize the observed response from thousands of measurements, there is still no commonly accepted explanation for these empirical formulae. This raises the theoretical questions: why are there two types of non-exponential behavior, and why are they so 'universal'?” Providing “orthogonal” insight to help address those basic questions provides a high-level impetus for the present work. Dispersive transport is often considered the underlying phenomenon responsible for the observation of dispersive kinetics in traditional (i.e., not ultrafast) conversions. The FokkerPlanck (FP) equation can be used to relate the probability density function ((pdf)) for Brownian motion in the presence of an external force field caused by a given potential surface. While in the absence of a potential, the mean-square displacement, Ær2æ, of the random walker evolves linearly with time, t, and thus, the Brownian random walker spreads according to Fick’s second law of diffusion to produce a Gaussian distribution (which is also a consequence of the central limit theorem, CLT); in cases of “anomalous” or dispersive diffusion, the normal distribution is not recovered. Instead, Ær2æ is related by the equation below Ær 2 æ t n
n 6¼ 1
ð1Þ
which is reflective of Levy-type transport statistics. In cases where 0 < n < 1, the processes are referred to as subdiffusive, while those where n > 1 are called superdiffusive. During subdiffusion, the random walker spreads less efficiently than a Brownian random walker, while the opposite is true for superdiffusion. Using the fractional FokkerPlanck (FFP) equation, which governs the temporal evolution of the (pdf) for subdiffusion in the presence of an external potential, Metzler and Klafter showed that the relaxation of single modes close to equilibrium obeys a MittagLeffler decay.8 Interestingly, the MittagLeffler behavior was shown to interpolate between an initial KWW function and a final inverse power law decay. For the subdiffusive OrnsteinUhlenbeck (OU) process (0 < n < 1), in which the restoring force on the particle is provided by a parabolic potential,
those workers have shown9 that the (pdf) exhibits asymmetry not observed in the Brownian case; the latter also relaxes faster to the thermal equilibrium (stationary) state defined by the (symmetrical, Gaussian) GibbsBoltzmann distribution, as compared to the OU process (which can be considered deceleratory, to use the terminology that will be introduced later). Note that the FFP equation is just one approach for modeling dispersive diffusion; alternatives include use of the asymmetric continuous time random walk (CTRW) model (discussed more below), generalized Langevin equations, or solution of the generalized master equation.9 Plonka has discussed7 that there are only three exact numerical values for n (1/3, 1/2, 2/3) for which there are known inverse Laplace transforms yielding stable distributions with moments. While “there is no problem with other numerical values...zero moments and the time-dependence of the rate coefficient for stretched exponential decay...were always seen as some disadvantages in this kind of interpretation of experimental results fitted with the Kolrausch relaxation function”.10 Unfortunately, as in cases of protein folding,11 there is often thought to be mechanistic ambiguity between seemingly equivalent fits to empirical data of both the stretched exponential and power law models.1 The Levy or “stable” distribution function is given by the following Fourier transform12 Z 1 ¥ ðizu jujm Þ Qm ðzÞ ¼ e du 0
