Reaction-Driven Nucleation Theory - The Journal of Physical

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C: Physical Processes in Nanomaterials and Nanostructures

Reaction-Driven Nucleation Theory Matt Wall, Brandi M. Cossairt, and Jonathan T.C. Liu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01368 • Publication Date (Web): 27 Mar 2018 Downloaded from http://pubs.acs.org on March 27, 2018

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Reaction-driven Nucleation Theory Matthew A. Wall1,†,*, Brandi M. Cossairt2, Jonathan T. C. Liu1 1

Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA

2

Department of Chemistry, University of Washington, Seattle, WA, 98195, USA

ABSTRACT: Understanding how nucleation is driven by chemical reactions is essential for the optimal design of materials grown from molecular precursors. Typically, the design and analysis of nucleation reactions is framed in terms of classical nucleation theory (CNT). However, the thermodynamic contribution of chemical reactions is not considered in CNT because it is formulated for pure phase transitions. We hypothesized that properly accounting for the thermodynamic influence of chemical reactions during nucleation would improve the accuracy of nucleation modeling with respect to CNT. Here we present reaction-driven nucleation theory (RDNT), based on a well-substantiated thermodynamic foundation that converges to the standard CNT in the special case of pure phase transitions without concomitant chemical reactions. The quantitative predictions of CNT and RDNT are compared for the representative cases of indium phosphide quantum dot (InP QD) nucleation and gold-nanoparticle (Au NP) nucleation. Under the conditions wherein InP QD and Au NP nucleation are known to occur, RDNT offers an excellent quantitative match to experimental observations, including the prediction of magic-sized clusters, while standard CNT incorrectly predicts that nucleation should be forbidden.

INTRODUCTION Nucleation is a ubiquitous phenomenon wherein a new thermodynamic phase or organized structure is born. Common examples of nucleation include the formation of raindrops from water vapor, carbon dioxide bubbles from champagne, and crystals from supersaturated solutions. In particular, the latter case has been of interest for researchers and various industries for decades due to the influence of crystal nucleation on, for example, the purity of chemical species, the proper functioning of oil and gas pipelines, and the effectiveness of pharmaceuticals.1-5 The advent of nanotechnology has further piqued interest in crystal nucleation, as the properties of nanocrystals, such as quantum dots and gold nanoparticles, depend strongly upon shape and size, which are largely determined by the nucleation process.6-12

ation theory (CNT).13, 15-19 In CNT, a thermodynamic analysis of atomic or molecular clusters as a function of size is used to identify the rate-limiting step in nucleation, such that the total expected time for the cluster to grow to a large size is assumed to be approximately equivalent to the expected time for the rate-limiting step to occur.15 This approach hypothesizes that the cluster with maximum formation energy will take the longest to form, and that once the cluster grows beyond that size, all subsequent growth will be favorable and much faster (Fig. 1). Despite being nearly a century old, CNT remains the most common approach for modeling nucleation processes.13 The widespread reliance on CNT is becoming increasingly criticized, however, as its predictions can deviate wildly – often by more than 10 orders of magnitude – from experimental observations.13, 20

The complexity of nucleation processes necessitates major approximations to be made for even the most advanced computational simulation methods.13 Because nucleation proceeds through the formation and evolution of clusters of atoms (or ions, molecules, etc.), with each cluster size being chemically distinct, an exhaustive chemical-kinetics analysis is precluded by the countless transient reactants and products involved. State-of-the-art molecular dynamics (MD) computations are limited by the time required to model the innumerable coalescence and dissolution events that precede and accompany nucleation.14 This computational burden becomes substantially more severe as the complexity of the nucleation system increases, such that the accurate modeling of common cases of nucleation driven by the chemical conversion of precursor molecules (e.g., AuCl2- to Au0) is prohibitive.

One notable oversight in CNT is that nucleation is modeled as a phase transition with no consideration of concomitant chemical reactions. Although the shorthand free-energy expressions in CNT are often written with abstract terms that could in principle be generalized, their quantitative forms are defined exclusively for phase transitions.15 This is particularly problematic for modern systems of interest, such as nanoparticle syntheses, which are driven by chemical reactions. In contrast to pure phase transformations like liquid water turning to ice, or lysozyme proteins assembling into crystals, reactiondriven nucleation often proceeds via solvated precursors undergoing reversible chemical reactions at cluster surfaces in order to become incorporated into growing clusters. Such chemical transformations between solvated and crystalline states are not properly accounted for within the current framework of nucleation theory.

The most popular route for simplifying the study of nucleation has been the use of what is now known as classical nucle-

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Given that a thermodynamic analysis of cluster formation is at the heart of CNT, we hypothesized that some of the observed inaccuracies may be corrected by deriving a thermodynamic model that properly describes the free energy contributions of chemical reactions during nucleation. Here we present a new thermodynamic framework for the free energy of cluster formation and apply it to the development of a reaction-driven nucleation theory (RDNT). The equations thus obtained are similar in form to those of CNT, but include terms that account for the free energy contribution from the chemical reaction system as a whole. In addition, unnecessary simplifications used by CNT to estimate the free energy of clusters are replaced by the best available computational tools, such as density functional theory (DFT).21-22 We demonstrate the utility of RDNT with the test case of indium phosphide (InP) quantum dot (QD) nucleation. This validation case is chosen because of the practical importance of InP QDs as a newly emerging alternative to toxic cadmiumbased QDs for applications in clean energy, electronics, and biomedical imaging, and also because this model system is known to exhibit a common property that complicates nucleation modeling.23-27 Specifically, InP nucleation involves the formation of magic sized clusters (MSCs) – particularly stable cluster geometries that result, for example, from low-energy electronic structures or favorable ligand coordination.28-29 MSCs are observed in many systems, but are not accounted for in CNT.28, 30-34 The RDNT framework not only allows for MSCs, but also illustrates that MSCs are an especially important feature for quantitative studies of nucleation. RESULTS The idea that a simple rate-limiting step can approximate a complex phenomenon is an exceptionally useful concept that is ubiquitous in theoretical modeling. In the case of nucleation, the rate-limiting step is identified by consideration of the work required to produce clusters of various sizes, n. Conceptually, the more work required for a cluster to form, the less likely it is to appear. This means that clusters with high work of formation will have low concentrations in the nucleation system, and thus their growth rates to larger sizes will be slow. For this reason, the cluster size with maximum work of formation (per growth unit), called the critical nucleus, n*, is expected to have the slowest rate of growing to a larger size. Thus, the rate-limiting step of nucleation will be the reaction between a monomer and the critical nucleus to achieve the cluster of the next size (i.e., n*+1). CNT takes this approach to determining the rate-limiting step of nucleation, but uses an expression for the work of cluster formation that we will show to be incorrect for systems of practical interest wherein nucleation is driven by chemical reactions. In order to develop a reaction-driven nucleation theory, we first had to derive a thermodynamically correct expression for the work of cluster formation, W(n). In contrast to CNT, which uses the free energy expressions of phase transformations to calculate the work of cluster formation, we begin our derivation with generalized free energy terms, and consider the abstract process of a set of reactants converting into a set of products, including the formation of a cluster growth unit. Although the equations obtained are not restricted to a particular phase transition or chemical reaction, here we describe the

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essential framework of RDNT in terms of nucleation from a solution without loss of generality. We denote the total change in free energy to convert one set of reactants to one set of products, which includes the generation of a cluster growth unit, by the shorthand !"#$%&'() . In order to obtain a desirable final form for the W(n) expression, we express !"#$%&'() as the sum of a bulk free energy term, !"*%$+ , and an excess free energy term, ,-./ 0/ 12: !"#$%&'() -./ 0/ 12 3 !"*%$+ -./ 12 4 ,-./ 0/ 12

(1)

The argument 0 represents the structural arrangement of cluster growth units, which can vary for a constant cluster size, n. The argument ! denotes the reaction coordinate. The reaction coordinate is a parameter that represents the completeness of a chemical reaction, such that the progression from reactants to products can be thought of as moving along the reaction coordinate. The bulk free energy term, !"*%$+ , is the change in free energy that results from converting one set of reactants to one set of products using the bulk crystallization stoichiometry. This is a subtle, but important point that will be discussed in greater detail later: the stoichiometry of a nucleation reaction tends to be different at the early stages of cluster formation than it is at the later stages (i.e., when n >> 1).35-36 The !"*%$+ term is calculated using the bulk stoichiometry corresponding to the balanced chemical reaction at equilibrium. The balanced chemical reaction of this bulk reaction is represented by an arbitrary set of reactants, Ri, with stoichiometric coefficients, ri, reacting to form arbitrary products, Pj, with stoichiometric coefficients, pj. By definition, at least one of the reactants in the nucleation reaction is a solvated species comprising some *%$+ or all components of the bulk crystalline monomer, 5-&2 . We denote this solvated species as MLx because it is a precursor to the crystalline monomer that may exist in a complex with arbitrary ligands, L. For example, AuCl2- is a solvated species that is the chemical precursor to the bulk crystalline monomer of gold nanoparticles (i.e., Au0). The balanced chemical reaction of bulk crystallization is thus given by: 67 87 4 69 89 4 : 4 5;< *%$+ = >7 ?7 4 >9 ?9 4 : 4 5-&2

(2)

Multicomponent reactions, wherein multiple solvated precursors generate a single crystalline growth unit (e.g., InP), follow the same general form as equation (2) (supporting information). The change in free energy that accompanies this reaction varies as the proportion of reactants and products in the system changes. Nonetheless, a simple expression for the free energy change per chemical reaction, given any concentration of reactants and products, may be formulated through the principles of chemical thermodynamics.37 This expression is defined in terms of a bulk reaction quotient, Qbulk, and an equilibrium constant, Keq: @*%$+

K

*%$+ GIH?I J L H5-&2 J A BCDE F Q GNM8N O)P M5;< O

(3)


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The Journal of Physical Chemistry R(S A BCDE F

K

GIH?I J L H5-&2J QT GNM8N O)P M5;< O

(S

(4)

Note that Keq is just the special case of Qbulk evaluated at equilibrium. The angular brackets in equations (3) and (4) denote thermodynamic activities, and k is Boltzmann’s constant. The expression for free energy change, U"VU1, in chemical thermodynamics is traditionally given at constant temperature and pressure (i.e., Gibbs free energy) and is written as a derivative with respect to the reaction coordinate, !. The free energy change with respect to producing one set of products from one set of reactants in the bulk crystallization reaction at any point in the reaction is given by: W

U"*%$+ @*%$+ \ X 3 BCDE [ U1 Y/Z R(S

(5)

In the standard thermodynamic treatment of crystallization reactions, it is assumed that the activity of the solid growth units, H5-&2 J, equals 1 (i.e., they are in a standard state). This means that all atoms in the clusters are treated as though they are inside of an infinitely large (i.e., bulk) crystal. However, the thermodynamic activity of real clusters is not equal to 1 because isolated clusters do not have the same total free energy as they would inside of an infinitely large crystal. In fact, real clusters often do not have the same structural properties as bulk crystals, and often do not even exhibit the same stoichiometry.29 These discrepancies necessitate a correction factor, which comes in the form of the excess cluster energy term, ,-./ 0/ 12. The excess cluster energy term is defined such that its addition to the bulk free energy term gives the exact change in free energy of the cluster formation reaction (i.e., equation (1) must be satisfied). When the cluster reaction has the same stoichiometry as the bulk crystallization reaction, ,-./ 0/ 12 is simply the energy of a cluster with n units minus the bulk energy of n units: ,-./ 0/ 12 3 "#$%&'() -./ 0/ 12 ] "*%$+ -./ 12^^^^^^ (6) -_`aB^bcdefgedhic6j2 Oftentimes, the stoichiometry of the cluster formation does not match that of the bulk crystallization process. In these cases, additional energy corrections must be included in ,-./ 0/ 12. An example of such a case is provided in the supporting information. We can now write the work of cluster formation, W(n), by multiplying the average (i.e., geometric mean) change in bulk kkkkkkk free energy per generated crystalline monomer, BCa.-@ *%$+ V R(S ^2, by the number of crystalline monomers in the cluster, n, then adding the excess energy term: kkkkkkk @ *%$+ (7) l-.2 3 .BCDE [ \ 4 ,-./ 0/ 12^ R(S Equation (7) is the central equation that enables thermodynamic simplification of the complex nucleation system – it is the work that must be put into a nucleation system in order to

generate a cluster of n atoms from the reactants involved in the nucleation reaction. The first term in equation (7) has bulk stoichiometry and can be determined from experimental measurements such as ultraviolet-visible (UV-Vis) absorbance spectroscopy.38 When the chemistry has been well described, the second term, ,-./ 0/ 12, can be calculated by means of common computational approaches, such as DFT, using the approach outlined herein. Equation (7) is similar in appearance to the work of cluster formation according to CNT: l-.2mnY 3 ].BCa. [

o \ 4 ,-.2mnY ^ o(S

(8)

The classical equation (8) differs from equation (7) in that the first term on the right-hand side depends only on the concentration, C, and solubility, Ceq, of solvated growth units (e.g., InP(solv)) and the second term uses a simplified expression of ,-./ 0/ 12 that treats clusters as having the same structural properties as a larger crystal and typically assumes that they are perfectly spherical. Moreover, the simplified excess cluster energy term requires that the cluster formation reaction has the same stoichiometry as the bulk crystallization reaction. Conceptual schematics of the work of cluster formation as treated by CNT and RDNT are provided in Figure 2. As a representative case, we investigated the qualitative and quantitative ability of CNT and RDNT to model the nucleation of InP QDs. The mechanism of cluster formation from InX3 and PY3 precursors has been previously investigated using AIMD simulations, and the findings have been corroborated by experimental syntheses of small InP clusters.29, 35 This provides a valuable opportunity to compare the utility of CNT and RDNT because critical validation data have been independently discovered. These studies report that molecular InX3 and PY3 precursors, where X and Y are arbitrary ligands, react to form clusters without ever producing bare In or P ions in solution. Experimental and theoretical investigations show that the surface of InP clusters are In-rich with a shell of X ligands. Specific cluster sizes (i.e., magic sized clusters, MSCs) are exceptionally stable, such that they can be experimentally isolated. For example, an MSC with the composition In37P20(O2CCH2Ph)51 has been characterized and shown to comprise an [In21P20]3+ core with an [In16(O2CCH2Ph)51]3shell, and an arrangement of atoms that deviates from the bulk phase structure of InP.29 A quantitative example of excess cluster energy and work of InP cluster formation according to CNT and RDNT is shown in Figure 3. The plots utilized the data reported by Cho et al. from hybrid DFT studies of InP clusters with hydrogen ligands.39 The clusters were constrained to have the bulk InP structure and stoichiometry, and therefore do not provide accurate descriptions of experimental InP clusters. Nonetheless, this data illustrates the main features of excess cluster energy and work of cluster formation. Neither the excess cluster energy nor the work of cluster formation demonstrate local minima when modeled by CNT (Fig. 3a, b). Accordingly, the occurrence of MSCs cannot be predicted by CNT. Note that the maximum work of cluster formation is greater than 80 kT for C/Ceq (i.e., [In][P]/ [In]eq[P]eq) less than 1055. According to CNT, a nucleation work barrier "80 kT energy forbids nuclea-



tion on an observable timescale.15 An argument can be proposed that free In and P ions are forming experimentally in concentrations too low to be detected, yet sufficient to favor nucleation. However, this is implausible given the exceedingly large activation energy barrier to the production of free In3+ ions from InX3.35 Consequently, sufficient concentrations of In3+ ions to exceed C/Ceq = 1055 are not feasible. Thus, CNT predicts that InP QD nucleation will not occur under the conditions in which nucleation is experimentally observed. In contrast to the predictions of CNT, the excess cluster energy and work of cluster formation modeled by RDNT (Fig. 3c,d) exhibit local minima at size m* (the magic sized cluster, MSC) and maxima at size n*, and have allowable nucleation work barriers (i.e., W(n*) - W(m*) 0.99) (Fig. 4b). The initial nucleation rate determined from curve-fitting to Ctot(t) with a known induction time (20 s)38 is J0 = 3.8 $ 10-4 M$s-1, which is a reasonable value for a nucleation rate under strong driving forces. The predicted mean InP cluster diameter is 3.19 nm, which is an excellent match to the experimentally observed mean of 3.2 nm (Fig. 4c).38 The predicted size distribution is narrower


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than what is experimentally observed, which is expected given the simplicity of our growth simulation. In particular, our input parameters are determined experimentally, and thus represent ensemble averages over many nucleation sites. Some of these sites will have reactant concentrations slightly greater or slightly lower than the bulk average, leading to statistical variation in parameters like #i and J0, which would widen the predicted size distribution. Additional considerations, such as Ostwald ripening, will affect the size distribution in some nucleation systems. We note, however, that the absorption maximum does not shift significantly once all precursors have been converted into clusters (i.e., between 20 and 80 minutes), which suggests that ripening is not a significant process in the later stages of the InP nucleation system investigated herein (Fig. S3). Nonetheless, future simulations will likely benefit from inclusion of ripening terms.40 DISCUSSION The utility of RDNT is not that it competes with advanced computational simulations, but that it interfaces with them to enable a multi-layer nucleation theory that can apply the molecular-level insights from DFT and MD to the entire nucleation process. The extension to CNT presented herein is necessary to enable this multi-layer approach, because the mechanisms of initial cluster formation revealed in DFT and MD simulations violate the assumptions of CNT (e.g., the capillarity approximation and bulk reaction stoichiometry fail). As we demonstrated for the case of InP QD nucleation, rigorous computations from first principles can be used to analyze the thermodynamics of molecular interactions (e.g., AIMD simulations) and cluster energetics (e.g., DFT), and then be fed into RDNT for accurate parameter selection. Once this training is complete (e.g., by defining ,-./ 0/ 12, etc.) quantitative predictions over long time scales can be performed at minimal computational cost to improve the understanding and synthetic design of nucleation reactions. We note that the application of the theory presented herein will be subject to the limitations of the computational methods used to estimate the excess cluster energy. In the case of InP quantum dots, we rely on the zerotemperature DFT results of Cho et al. that do not account for entropic contributions. The detailed mechanism of InP cluster formation and the appearance of MSCs offer important qualitative tests for nucleation theories. Because CNT assumes that all cluster sizes have the same stoichiometry and bonding symmetry as a bulk phase crystal (capillarity approximation), CNT cannot model the excess energy of non-stoichiometric, non-bulk phase InP clusters. Moreover, MSCs are not observed in CNT because the excess cluster energy is given as a function of surface area wherein local minima are precluded. In contrast, RDNT models the excess cluster energy exactly, even in nonstoichiometric clusters without bulk-phase structure. Because certain combinations of size and geometry exhibit local minima in the work of cluster formation, RDNT can identify MSCs and can successfully model excess free energies during cluster formations. The driving force for InP nucleation again shows disparity between CNT and RDNT. In CNT, a driving force for nucleation is established when solute molecules accumulate in concentrations too high to be dissolved at equilibrium (i.e., [In3+] >> [In3+]eq, [P3-] >> [P3-]eq), such that a phase transformation

becomes favorable. It is important to note that cluster formation proceeds from solvated to crystalline phase without chemical reactions in CNT, so In3+ and P3- ions would have to be generated as intermediates that subsequently cluster together. This disagrees with the clustering mechanism observed in AIMD simulations, wherein bare In3+ and P3- ions are not observed.35 Thus, even if the solubility of In3+ and P3- ions is vanishingly small, the concentration of In3+ ions in solution is effectively zero and cannot explain the driving force for InP nucleation. On the other hand, the driving force of nucleation in RDNT is the decrease in free energy that accompanies the chemical conversion of reactants to products. Thus, the thermodynamic barrier to nucleation can be overcome by sufficiently exergonic chemical reactions accompanying the formation of cluster growth units. Therefore, RDNT offers a seamless qualitative fit to the known InP data, whereas CNT fails on all accounts. This statement applies beyond the case of InP nucleation. For example, DFT studies demonstrate that gold nucleation most likely proceeds directly from AuCl2monomers reacting with other AuxCly species (where x may or may not equal y) while never producing solvated Au0 monomers. RDNT allows for this, while CNT does not. In the event that CNT proves successful in describing a nucleation system, it must also be true that RDNT successfully models the system. This is because CNT is just a special case of RDNT. In particular, CNT naturally arises from RDNT when the nucleation reaction is a pure phase transition between a solvated and crystalline state, o&y$z = o#){&' , and the excess cluster energy is described with bulk stoichiometry and bulk phase structure, ,-./ 0/ 12 A ,-.2mnY . For such a reaction, @*%$+ 3 |Vo&y$z, which leads to .BCDEu@*%$+ VR(S v 3 ].BCDEuo&y$z Vo(S v. In this simple case, the work of cluster formation is given by the classical expression: .BCDE-@*%$+ V This R(S ^2 4 ,-./ 0/ 12 3 ].BCDE-o&y$z Vo(S ^2 4 ,-.2mnY . common form of l-./ 0/ 12 assumes constant supersaturation, but variable supersaturation can be accounted for by replacing the concentration and reaction quotient with their geometric means. We note that improvements to nucleation theory have recently been made by replacing the oversimplified classical nucleation mechanism – wherein solvated monomers directly coalesce into clusters with bulk crystalline properties – with a multistep hypothesis that better matches experimental observations.41-46 According to multistep nucleation theory, a solution first separates into solute-poor and solute-rich regions, whereupon nucleation proceeds within the solute-rich regions. Oftentimes an initial cluster forms that does not have the same crystalline structure as the bulk crystal. Continued growth and structural rearrangement of this cluster eventually yields a crystalline form that has similar characteristics to the material at larger sizes. Multistep nucleation has a different rate law47 than CNT, but still characterizes nucleation as having a critical nucleus size – that is, a representative rate-limiting step that can be identified by a thermodynamic analysis of the system.1 The improved thermodynamic foundation of RDNT not only allows for the possibility of multistep mechanisms, but extends the reach of multistep nucleation theory to increasingly complex systems. For systems exhibiting multistep nucleation, the generalized work of cluster formation (7) should be evaluated under the conditions inside of the solute-rich regions (e.g., higher reactant concentration, etc.). The approximation



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given in (9) is applicable to multistep nucleation, and the exact expression is provided in the supporting information. The results presented here demonstrate that RDNT offers a powerful extension to CNT for modeling nucleation driven by chemical reactions, but RDNT does have limitations. The central hypothesis of RDNT (and CNT) is that the rate at which very large clusters (i.e., crystals) appear is approximated by a rate-limiting step when a critical nucleus, n*, can be identified. However, if the work of cluster formation is monotonically decreasing, then there is never a point at which increasing the cluster size requires the input of free energy, and thermodynamic analyses of cluster sizes cannot identify a kinetic ratelimiting step. This scenario is typically referred to as spinodal decomposition, and is most likely to occur for systems driven by chemical reactions wherein nucleation is strongly favored and no MSCs exist.11 In these cases, however, nucleation is very rapid and likely to fall in the range computationally accessible by MD simulations, which are preferable whenever possible. In general, when ,-./ 0/ 12 is known from DFT, the values of Keq that offer the most reaction control can be bookended by determining the value that yields W(n*) – W(m*) = 80 kT (lower bound) and the value that first makes l-./ 0/ 12 monotonically decreasing (upper bound). CONCLUSION We have developed a nucleation theory derived from chemical thermodynamics that quantitatively accounts for the role of chemical reactions in addition to phase transformation during nucleation. When nucleation occurs as a pure phase transformation without concomitant chemical reactions, and the excess cluster energy is approximated by a simplified approximation, ,-.2mnY , the classical equations of CNT naturally arise from our theoretical framework. The importance of our holistic approach to modeling nucleation is evidenced in the important cases of InP QD and gold nanoparticle syntheses, wherein nucleation is forbidden by CNT, but favored by RDNT under the conditions where nucleation is experimentally observed. Moreover, the quantitative predictions of RDNT are in excellent agreement with the experimental observations of InP QD nucleation. We envision that RDNT can serve as a critical component of future multiscale approaches to nucleation modeling. For the example case of InP nucleation, DFT computations reveal the excess energy of different cluster sizes, MD simulations identify the thermodynamics and mechanism of reactions between precursor species, and RDNT has its parameter set trained by these DFT and MD results. RDNT can then provide otherwise unattainable quantitative analyses and predictions of nucleation systems over long time periods at low computational cost. Such a multiscale framework will move the field closer to achieving complete computational design of desired nucleation products such as precisely controlled nanoparticle morphologies and optimal drug polymorphs.


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Table 1. Nucleation parameters for InP QD formation. Fixed inputs Variable

#i (s)

n* (InP units)

J0 (M$s-1)

52 52

Measurement 10 – 30 Theory

Fitted parameters

20

katt -1 -1

Prediction

(M $s )

Fit to UV-Vis

Mean Size (nm)

_

_

N/A

3.2

3.8 $ 10-4

0.49

R2 >0.99

3.19

The experimental values of #i, C(t), and mean size were obtained from Gary et al.38 and the experimental critical nucleus was estimated by adding the difference in critical nucleus and magic cluster size predicted by Cho et al.39 to the experimental magic cluster size reported in Gary et al.29 3.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Derivation of reaction-driven nucleation theory and supporting figures (PDF)

AUTHOR INFORMATION Corresponding Author * [email protected]

Present Addresses †Institute for Systems Biology, Seattle, WA, 98109

Author Contributions M.A.W, conceived the theory, derived the theoretical framework, performed all simulations, and wrote the paper. B.M.C oversaw the application of the theory to the test case of Indium Phosphide quantum dots and wrote the paper. J.T.C.L supervised the research and wrote the paper.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT The authors acknowledge support from the NIH/NIBIB – R21 EB015016 (J.T.C.L) and the department of mechanical engineering at the University of Washington. BMC would like to thank the David and Lucile Packard Foundation and the National Science Foundation (CHE 1552164) for support of research that contributed to this study.

ABBREVIATIONS CNT, classical nucleation theory; RDNT, reaction-driven nucleation theory.

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Reaction-Driven Nucleation Theory - The Journal of Physical

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