In LaMer burst nucleation, the individual nucleation events happen en masse, quasi-simultaneously, and at nearly identical homogeneous conditions. These properties make LaMer burst nucleation important for applications that require monodispersed particles and also for theoretical analyses. Sugimoto and co-workers predicted that the number of nuclei generated during a LaMer burst depends only on the solute supply rate and the growth rate, independent of the nucleation kinetics. Some experiments confirm that solute supply kinetics control the number of nuclei, but flaws in the original theoretical analysis raise questions about the predicted roles of growth and nucleation kinetics. We provide a rigorous analysis of the coupled equations that govern concentrations of nuclei and solutes. Our analysis confirms that the number of nuclei is largely determined by the solute supply and growth rates, but our predicted relationship differs from that of Sugimoto et al. Moreover, we find that additional nucleus size dependent corrections should emerge in systems with slow growth kinetics. Finally, we show how the nucleation kinetics determine the particle size distribution. We suggest that measured particle size distributions might therefore provide ways to test theoretical models of homogeneous nucleation kinetics.

Recent theories and simulations suggest that molecular additives can bind to the surfaces of nuclei, lower the surface energy, and accelerate nucleation. Experiments have shown that oligomeric and polymeric additives can also modify nucleation rates of proteins, ice, and minerals; however, general design principles for oligomeric or polymeric promoters do not yet exist. Here we investigate oligomeric additives for which each segment of the oligomer can bind to surfaces of nuclei. We use semigrand canonical Monte Carlo simulations in a Potts lattice gas model to study the effects of oligomer chain length, volume fraction, and binding strength. We find that increasing each of those parameters lowers the nucleation barrier. At extremely low oligomer concentrations, the nucleation kinetics can be modeled as though each oligomer is a heterogeneous nucleation site in solution.

Existing methods to compute free-energy differences between polymorphs use harmonic approximations, advanced non-Boltzmann bias sampling techniques, and/or multistage free-energy perturbations. This work demonstrates how Bennett’s diabat interpolation method (J. Comput. Phys.1976, 22, 245) can be combined with energy gaps from lattice-switch Monte Carlo techniques (Phys. Rev. E2000, 61, 906) to swiftly estimate polymorph free-energy differences. The new method requires only two unbiased molecular dynamics simulations, one for each polymorph. To illustrate the new method, we compute the free-energy difference between face-centered cubic and body-centered cubic polymorphs for a Gaussian core solid. We discuss the justification for parabolic models of the free-energy diabats and similarities to methods that have been used in studies of electron transfer.

In ethylene polymerization by the Phillips catalyst, inorganic Cr(II) sites are believed to be activated by reaction with ethylene to form (alkyl)CrIII sites, in a process that takes about 1 h at ca. 373 K. The detailed mechanism of this spontaneous self-initiation has long remained unknown. It must account both for the formation of the first Cr–C bond and for the one-electron oxidation of Cr(II) to Cr(III). In this study, we used density functional theory to investigate a two-step initiation mechanism by which ethylene oxidative addition leads first to various (organo)CrIV sites, and subsequent Cr–C bond homolysis gives (organo)CrIII sites capable of polymerizing ethylene. Pathways involving spin crossing, C–H oxidative addition, H atom transfer, and Cr–C bond homolytic cleavage were explored using a chromasiloxane cluster model. In particular, we used classical variational transition theory to compute free energy barriers and estimate rates for bond homolysis. A viable route to a four-coordinate bis(alkyl)CrIV site was found via spin crossing in a bis(ethylene)CrII complex followed by intramolecular H atom transfer. However, the barrier for subsequent Cr–C bond homolysis is a formidable 209 kJ/mol. Increasing the Cr coordination number to 6 with additional siloxane ligands lowers the homolysis barrier to just 47 kJ/mol, similar to reported homolysis paths in molecular [CrR(H2O)53+] complexes. However, siloxane coordination also raises the barrier for the prior oxidative addition step to form the bis(alkyl)CrIV site. Thus, we suggest that hemilability in the silica “ligand” may facilitate the homolysis step while still allowing the oxidative addition of ethylene.Keywords: bond homolysis; chromium; density functional theory; ethylene polymerization; initiation mechanism; Phillips catalyst

We develop a theory to account for variable coverage of trace additives that lower the interfacial free energy for nucleation. The free energy landscape is based on classical nucleation theory and a statistical mechanical model for Langmuir adsorption. Dynamics are modeled by diffusion-controlled attachment and detachment of solutes and adsorbing additives. We compare the mechanism and kinetics from a mean-field model, a projection of the dynamics and free energy surface onto nucleus size, and a full two-dimensional calculation using Kramers–Langer–Berezhkovskii–Szabo theory. The fluctuating coverage model predicts rates more accurately than mean-field models of the same process primarily because it more accurately estimates the potential of mean force along the size coordinate.

Olefin epoxidation catalyzed by methyltrioxorhenium (MTO, CH3ReO3) is strongly accelerated in the presence of H2O. The participation of H2O in each of the elementary steps of the catalytic cycle, involving the formation of the peroxo complexes (CH3ReO2(η2-O2), A, and CH3ReO(η2-O2)2(H2O), B), as well as in their subsequent epoxidation of cyclohexene, was examined in aqueous acetonitrile. Experimental measurements demonstrate that the epoxidation steps exhibit only weak [H2O] dependence, attributed by DFT calculations to hydrogen bonding between uncoordinated H2O and a peroxo ligand. The primary cause of the observed H2O acceleration is the strong co-catalytic effect of water on the rates at which A and B are regenerated and consequently on the relative abundances of the three interconverting Re-containing species at steady state. Proton transfer from weakly coordinated H2O2 to the oxo ligands of MTO and A, resulting in peroxo complex formation, is directly mediated by solvent H2O molecules. Computed activation parameters and kinetic isotope effects, in combination with proton-inventory experiments, suggest a proton shuttle involving one or (most favorably) two H2O molecules in the key ligand-exchange steps to form A and B from MTO and A, respectively.

Nucleation and crystal growth are important in material synthesis, climate modeling, biomineralization, and pharmaceutical formulation. Despite tremendous efforts, the mechanisms and kinetics of nucleation remain elusive to both theory and experiment. Here we investigate sodium chloride (NaCl) nucleation from supersaturated brines using seeded atomistic simulations, polymorph-specific order parameters, and elements of classical nucleation theory. We find that NaCl nucleates via the common rock salt structure. Ion desolvation—not diffusion—is identified as the limiting resistance to attachment. Two different analyses give approximately consistent attachment kinetics: diffusion along the nucleus size coordinate and reaction-diffusion analysis of approach-to-coexistence simulation data from Aragones et al. (J. Chem. Phys. 2012, 136, 244508). Our simulations were performed at realistic supersaturations to enable the first direct comparison to experimental nucleation rates for this system. The computed and measured rates converge to a common upper limit at extremely high supersaturation. However, our rate predictions are between 15 and 30 orders of magnitude too fast. We comment on possible origins of the large discrepancy.

The mechanism of ethylene polymerization by the widely used Phillips catalyst remains controversial. In this work, we compare initiation, propagation, and termination pathways computationally using small chromasiloxane cluster models for several previously proposed and new mechanisms. Where possible, we consider complete catalytic cycles and compare predicted kinetics, active site abundances, and polymer molecular weights to known properties of the Phillips catalyst. Prohibitively high activation barriers for propagation rule out previously proposed chromacycle ring expansion and Green–Rooney (alternating alkylidene/chromacycle) mechanisms. A new oxachromacycle ring expansion mechanism has a plausible propagation barrier, but initiation is prohibitively slow. On sites with adjacent bridging hydroxyls, either ≡Si(OH)CrII-alkyl or ≡Si(OH)CrIII-alkyl, initiated by proton transfer from ethylene, chain growth by a Cossee–Arlman-type mechanism is fast. However, the initiation step is uphill and extremely slow, so essentially all sites remain trapped in a dormant state. In addition, these sites make only oligomers because when all pathways are considered, termination is faster than propagation. A monoalkylchromium(III) site without an adjacent proton, (≡SiO)2Cr-alkyl, is viable as an active site for polymerization, although its precise origin remains unknown.Keywords: chain termination; chromium; density functional theory; ethylene polymerization; initiation; Phillips catalyst

We present new algorithms for conducting transition path sampling (TPS). Permutation shooting rigorously preserves the total energy and momentum of the initial trajectory and is simple to implement even for rigid water molecules. Versions of aimless shooting and permutation shooting that use flexible-length trajectories have simple acceptance criteria and are more computationally efficient than fixed-length versions. Flexible-length permutation shooting and inertial likelihood maximization are used to identify the reaction coordinate for vacancy migration in a two-dimensional trigonal crystal of Lennard-Jones particles. The optimized reaction coordinate eliminates nearly all recrossing of the transition state dividing surface.

We examine the capabilities and foundations of three landmark rate theories: harmonic transition state theory, classical nucleation theory, and the Marcus theory of electron transfer. Each of the three classic rate theories is widely used to predict rates and trends. They are also used “in reverse” to interpret experimental data with no computation at all. Their common foundations include a quasi-equilibrium assumption and dimensionality reduction to a physically meaningful, one-dimensional, and broadly applicable reaction coordinate. Many applications lie beyond the scope of the classic theories, so rare events research has pursued trajectory-based methods that efficiently predict accurate rate constants even when the reaction coordinate and mechanistic details are unknown. Trajectory based rare events methods achieved these ambitious goals, but (by construction) they provide rates rather than mechanistic understanding. We briefly discuss recent efforts to identify reaction coordinates, including methods which provide abstract statistically defined coordinates and those which identify physical collective variables. Finally, we note some natural synergies between existing simulation methods which might help discover simple and powerful quasi-equilibrium theories for the many applications that fall beyond the scope of the classic rate theories.

From a hypothetical perfect dividing surface, all trajectories commit to opposite basins in forward and backward time without recrossing, transition state theory is exact, the transmission coefficient is one, and the committor distribution is perfectly focused at 1/2. However, chemical reactions in solution and other real systems often have dynamical trajectories that recross the dividing surface. To separate true dynamical effects from effects of a nonoptimal dividing surface, the dividing surface and/or reaction coordinate should be optimized before computing transmission coefficients. For NaCl dissociation in TIP3P water, we show that recrossing persists even when the 1/2-committor surface itself is used as the dividing surface, providing evidence that recrossing cannot be fully eliminated from the dynamics for any configurational coordinate. Consistent with this finding, inertial likelihood maximization finds a combination of ion-pair distance and two solvent coordinates that improves the committor distribution and increases the transmission coefficient relative to those for ion-pair distance alone, but recrossing is not entirely eliminated. Free energy surfaces for the coordinates identified by inertial likelihood maximization show that the intrinsic recrossing stems from anharmonicity and shallow intermediates that remain after dimensionality reduction to the dynamically important variables.

The formation of peroxorhenium complexes by activation of H2O2 is key in selective oxidation reactions catalyzed by CH3ReO3 (methyltrioxorhenium, MTO). Previous reports on the thermodynamics and kinetics of these reactions are inconsistent with each other and sometimes internally inconsistent. New experiments and calculations using density functional theory with the ωB97X-D and augmented def2-TZVP basis sets were conducted to better understand these reactions and to provide a strong experimental foundation for benchmarking computational studies involving MTO and its derivatives. Including solvation contributions to the free energies as well as tunneling corrections, we compute negative reaction enthalpies for each reaction and correctly predict the hydration state of all complexes in aqueous CH3CN. New rate constants for each of the forward and reverse reactions were both measured and computed as a function of temperature, providing a complete set of consistent activation parameters. New, independent measurements of equilibrium constants do not indicate strong cooperativity in peroxide ligand binding, as was previously reported. The free energy barriers for formation of both CH3ReO2(η2-O2) (A) and CH3ReO(η2-O2)2(H2O) (B) are predominantly entropic, and the former is much smaller than a previously reported value. Computed rate constants for a direct ligand-exchange mechanism, and for a mechanism in which a water molecule facilitates ligand-exchange via proton transfer in the transition state, differ by at least 7 orders of magnitude. The latter, water-assisted mechanism is predicted to be much faster and is consequently in much closer agreement with the experimentally measured kinetics. Experiments confirm the predicted catalytic role of water: the kinetics of both steps are strongly dependent on the water concentration, and water appears directly in the rate law.

Nucleation kinetics, induction times, and metastable zone widths are often modeled in quiescent, quasi-steady, and/or spatially uniform concentration fields. However, freezing an aqueous solution can concentrate the solute and effectively increase the supersaturation. During the freezing process, a boundary layer of giant supersaturation develops ahead of the moving ice front. We develop stochastic models of nucleation in the boundary layer when the growing ice perfectly excludes the solute for a one-dimensional system. The models make three simplifying assumptions: quasi-stationary nucleation kinetics, nuclei that are small compared to the boundary layer thickness, and a constant solvent crystallization growth velocity. Whether heterogeneous on the ice surface, or homogeneous in the boundary layer, the models suggest that nucleation is dramatically accelerated by the growing ice. For methane hydrates, which form at conditions similar to that of ice, induction times for hydrate nucleation can be reduced by as much as 10105 times because of the moving supersaturation zone.

Classical nucleation theory is notoriously inaccurate when using the macroscopic surface free energy for a planar interface. We examine the size dependence of the surface free energy for TIP4P/2005 water nanodroplets (radii ranging from 0.7 to 1.6 nm) at 300 K with the mitosis method, that is, by reversibly splitting the droplets into two subclusters. We calculate the Tolman length to be −0.56 ± 0.09 Å, which indicates that the surface free energy of water droplets that we investigated is 5–11 mJ/m2 greater than the planar surface free energy. We incorporate the computed Tolman length into a modified classical nucleation theory (δ-CNT) and obtain modified expressions for the critical nucleus size and barrier height. δ-CNT leads to excellent agreement with independently measured nucleation kinetics.Keywords: classical nucleation theory; curvature; hybrid Monte Carlo; mitosis method; surface tension;

Methane hydrates are ice-like inclusion compounds with importance to the oil and natural gas industry, global climate change, and gas transportation and storage. The molecular mechanism by which these compounds form under conditions relevant to industry and nature remains mysterious. To understand the mechanism of methane hydrate nucleation from supersaturated aqueous solutions, we performed simulations at controlled and realistic supersaturation. We found that critical nuclei are extremely large and that homogeneous nucleation rates are extremely low. Our findings suggest that nucleation of methane hydrates under these realistic conditions cannot occur by a homogeneous mechanism.

For bilinearly coupled oscillator models, we examine the statistical relationship between transmission coefficients and committor distribution variances for reaction coordinates obtained by likelihood maximization. Transmission coefficients usually but not always increase as committor distributions narrow for the original version of likelihood maximization. We propose a new inertial version of likelihood maximization that uses velocity information to optimize purely configuration dependent coordinates. The coordinates from inertial likelihood maximization have higher transmission coefficients than coordinates from the original likelihood maximization procedure. Inertial likelihood maximization should be useful for understanding mechanisms of inertial reactions from atomistic simulations.Graphical abstractHighlights► Original likelihood maximization procedures may give low transmission coefficients. ► Inertial likelihood maximization coordinates have high transmission coefficients. ► Inertial likelihood maximization reverts to original method for diffusive dynamics.

The metastability limit can be defined as the average supersaturation at which nucleation first occurs when the supersaturation steadily increases under isothermal conditions. We present a quasi-steady stochastic model for such experiments in terms of the nucleation free energy barrier at some reference supersaturation and in terms of a Damkohler number involving the kinetic prefactor, the observation volume, and the supersaturation schedule as a function of time. Classical nucleation theory provides a model for the dependence of the nucleation free energy barrier on the slowly increasing supersaturation. We derive the average critical supersaturation as a function of two dimensionless parameters and the type of supersaturation schedule. For slow supersaturation rates, the metastable zone width expression for one schedule collapses to earlier expressions from Volmer, Kashchiev, Verdoes, and van Rosmalen, which arose from alternative definitions of the metastability limit. Our findings show that isothermal metastable zone width experiments cannot be performed slowly enough to attain the limit where homogeneous nucleation occurs at the equilibrium boundary of the metastable zone. We also suggest a simple linear regression strategy to extract kinetic parameters for nucleation from isothermal metastability limit experiments.