Nucleation and Growth Kinetics from LaMer Burst Data - The Journal

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Nucleation and Growth Kinetics from LaMer Burst Data Daniel B. K. Chu, Jonathan S Owen, and Baron Peters J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b08368 • Publication Date (Web): 20 Sep 2017 Downloaded from http://pubs.acs.org on September 24, 2017

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The Journal of Physical Chemistry

Nucleation and Growth Kinetics from LaMer Burst Data Daniel B. K. Chu,1 Jonathan S. Owen,2 Baron Peters1,3* 1Department 2Department

of Chemical Engineering, University of California, Santa Barbara, CA 93106 of Chemistry, Columbia University, New York, NY 10027

3Department

of Chemistry and Biochemistry, University of California, Santa Barbara, CA 93106

ABSTRACT: 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 coworkers 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.

INTRODUCTION In most nucleation processes, the kinetics are extremely difficult to model because nucleation occurs heterogeneously at sites with unknown characteristics.1-2 In contrast, the in situ source of supersaturation in a LaMer burst nucleation process yields a massive population of nuclei all born simultaneously, under identical conditions, and most likely by homogeneous nucleation. As first understood by LaMer and Dinegar,3-4 a steady supply of solutes initially drives supersaturation toward the metastable zone limit. Near the edge of the metastable zone, the nucleation rate skyrockets and a large population of nuclei are almost simultaneously born. As these nuclei begin to grow, they drive the supersaturation downward again and thereby arrest all further nucleation. The distinct stages of a LaMer burst are shown above in the table of contents figure. In the figure, C(t) is the solute concentration, and Cm is the maximum in C(t), attained at the time tm. Many experiments4-7 qualitatively confirm the behavior in the table of contents figure, and the separate nucleation and growth stages of a LaMer burst are a widely recognized route to monodisperse particles.8-11 However, quantitative predictions of the number, size, and polydispersity in the particle size distribution (PSD) face many complications. In some experiments, the injection of solutes or reducing agents creates a supersaturated plume where nuclei can form before nucleation begins in the bulk solution.5, 10 Even under the most carefully controlled conditions, aggregation and ripening processes begin to modify the PSD shortly after the nucleation burst.12-13 If these

complications can be mitigated, one could exploit the naturally controlled conditions during a LaMer burst to develop accurate models for the nucleation and growth kinetics. Recent advances have added new ways to generate, analyze, and model the data from these complex processes. In particular, we mention three experimental techniques: (i) New time-resolved pair-distribution techniques can track the PSD even during the short nucleation burst stages.14-17 (ii) New syntheses use precursors that decompose with tunable kinetics15, 17-19 to control the in situ homogeneous solute supply rate. (iii) For semiconductor quantum dots, electron transfer and Forster resonance energy transfer properties can be used to characterize the thickness, density, and composition of the organic ligands.20 These experiments provide opportunities to synthesize monodisperse nanoparticles with tunable properties under precisely controlled conditions and to probe the PSD before aggregation and ripening processes can occur. Unlike processes that are dominated by heterogeneous and/or secondary nucleation, the kinetics of LaMer burst nucleation can be precisely and reliably controlled through the rate of the solute generation reaction, e.g. by adjusting the solvent pH, ionic strength, temperature, solute precursor ligands, etc.8 The well-defined nature of the conditions during homogeneous nucleation and the wealth of kinetic trends that can be examined make LaMer burst nucleation ideal for theoretical studies. Thus far, LaMer burst studies have focused on inorganic nanoparticle syntheses, but similar ideas will surely impact future efforts to create mono-

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disperse nanocrystalline pharmaceuticals,21 food products,22 and perhaps building blocks for growing larger crystals by oriented attachment.23 Current models of LaMer burst nucleation span several levels of detail: stochastic population balance models,24-25 deterministic species balance models,7, 26-27 and models that include additional ripening and aggregation processes.25 Sugimoto introduced an extremely simple model of LaMer burst nucleation that is readily used for interpreting experiments.8, 28 This paper examines the assumptions and analysis behind the Sugimoto model to motivate certain improvements to its accuracy, versatility, and predictive power. THE MODEL OF SUGIMOTO Sugimoto’s starting point is a pair of differential equations that govern the concentration of solutes C(t) and the population of nuclei pT(t):

dC = Q − J n * −GpT dt

(1)

and

dpT / dt = J

(2)

Here n* is the critical nucleus size, J is the nucleation rate, pT is the number of post-critical nuclei per volume, G is the growth rate in solutes attached per time per nucleus, and Q is the solute supply rate per volume. Equation (1) is a species balance equation for the solutes and equation (2) is the definition of the nucleation rate.8, 28 To simplify the analysis of events happening during the nucleation period, Sugimoto assumes (i) that G is a constant, and (ii) that dC/dt vanishes. Using these assumptions, Sugimoto combines equations (1) and (2) yielding his nucleation balance equation28

Q = n * dpT / dt + GpT

(3)

In Sugimoto’s analysis, Q and n* are also treated as constants. Sugimoto then solves equation (3) as an ODE for t > tm, where tm is the time of maximum solute concentration as shown in Figure 1. For the initial conditions, he assumes that nucleation begins at the time tm, i.e. that pT(tm) = 0.28 Equation (3) is linear with constant coefficients, so Sugimoto obtains the number of nuclei as a function of time as

pT =

Q (1 − exp [ −(t − tm )G / n *]) for t ≥ tm G

(4)

The total number of nuclei generated during the burst is

pT∞ = Q / G

(5)

where the superscript “∞” indicates a time much longer than n*/G, but still short compared to the duration of the growth process. Equation (5) is remarkable because (i) it

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predicts that the total number of nuclei does not depend on any aspect of the nucleation kinetics, and (ii) it predicts a linearly increasing relationship between the Q and

pT∞ .

These key predictions have been corroborated by several experiments on AgCl and AgBr nucleation.8, 28-29 Experiments on chalcogenides also show an increasing relationship between G and

pT∞ , but there are significant depar-

tures from the linear relationship in equation (5).15, 17, 19 Sugimoto and others have used equation (5) to infer the growth rates, but as shown below certain problems in the derivation of equation (5) suggest that the observed slope is not actually 1/G. EXTENSION OF THE SUGIMOTO MODEL Several aspects of the model by Sugimoto et al. can potentially be improved. An obvious point of concern is the assumption of a size and concentration independent growth rate G. The concentration dependence of G may be negligible because the concentration changes are small during the nucleation stage, but the size dependence of G is probably important. When nuclei are still near their critical size n* they grow by random attachment and detachment events with very little bias toward attachment.2, 30 As the nuclei become much larger than their critical size the driving force for growth approaches its macroscopic limit, but growth rates may still depend on size through diffusion controlled attachment kinetics (where the boundary layer thickness is proportional to the particle radius) or through surface reaction limited attachment (where the rate of attachment is proportional to the particle surface area).2 More accurate models with size dependent growth rates can be formulated, but they are not as easily solved and used to analyze experimental data. Therefore, we also assume a concentration and size independent attachment frequency G throughout this article. This article reexamines Sugimoto’s analyses leading to equations (3) – (5), i.e. to the prediction that the nucleation rate dpT/dt is entirely determined by Q, G, and n*, and that pT∞ is entirely determined by Q and G. Equations (3) - (5) warrant reexamination for two reasons: (i) The counterintuitive irrelevance of the nucleation kinetics to the number of particles (and therefore their mean size), as predicted by equation (5), has been verified by Sugimoto’s experiments.28-29, 31 (ii) While experiments can directly confirm the depend∞

ence of pT on Q, flaws in the derivation of equations (3) – (5) raise questions as to whether the slope actually corresponds to 1/G. Equation (3) is true at and only at t = tm, but Sugimoto et al. used and solved equation (3) as though it remains valid for all t ≥ tm. In fact, equation (3) is not a differential equation. It is a condition on the solutions of the coupled differential equations (1) and (2), much like an initial condition. Additionally, the initial condition invoked by Sugimoto, pT(tm) = 0, is incompatible with any nucleation rate expression that depends monotonically on supersaturation. For all such models, nucleation would begin slightly before tm, climax at tm, and then decline toward zero again after tm.


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Here we provide a more rigorous analysis starting from the equations (1) and (2). Clearly, dJ/dt = 0 at t = tm. Thus equation (3) remains valid, but only as a condition on the solutions, not as an ordinary differential equation. A second condition is required to solve the two coupled first order equations. Differentiating equation (1) again and using J = dpT/dt gives

also attained at t = tm. Both Jm and Cm are unknown parameters to be determined (or eliminated). J is an increasing function of the concentration C, but we can also think of J as a function of time (see Figure 1) because C depends on time. Using equation (6) to compute d2J/dt2 gives

d 2J dJ = −GJ dt 2 dC

2

d C = −G J dt 2

at t = tm

(6)

Now equations (3) and (6) are constraints that must be imposed on the solutions of equations (1) and (2). Figure 1 summarizes the equations that describe each portion of the LaMer burst nucleation process.

at t = tm

(8)

J has a strong dependence on concentration. Near the time tm the maximum in C(t) leads to a sharp spike in the nucleation rate as depicted in Figure 1. For these reasons (and for additional reasons described below) the time dependence of J is more difficult to model than that of lnJ. Thus we rewrite equation (8) as

d2J GJ 2 d ln J = − at t = tm dt 2 C d ln C

(9)

Nucleation kinetics are somewhat like those of a chemical reaction with high reaction order. For nuclei that form via a sharply defined critical-sized cluster n*, e.g. a transition state between pre- and post-critical magic clusters, one expects a nucleation rate expression of elementary molecular kinetics form, J = kCn*. Such rate expressions yield d ln J / d ln C = n * For nuclei that form via the classical model,33 the nucleation rate expression is J = Aexp[B/ln2(C/Csat)] where B depends on the temperature, the interfacial free energy, and the nucleus shape. The classical nucleation theory also yields d ln J / d ln C = nm* , as is well-known from the nucleation theorem.34 However, the classical theory predicts nm* = 2 B / ln 3 (C / Csat ) , i.e. the

Figure 1: Each stage of a LaMer burst nucleation process is described by different equations that relate the solute concentration C, the nucleation rate J, the number of nuclei pT, the growth rate G, and the nucleus size n*.

critical nucleus size now depends on concentration. For these two nucleation theories and others, a LaMer burst model formulated in terms of dlnJ/dlnC will help to extract the critical nucleus size from experimental data. In all subsequent expressions, the postfix |m indicates that the preceeding quantity should be evaluated at the conditions of point t = tm. Also in all subsequent formulas, (dlnJ/dlnC)|m will be replaced with

nm* .

To enable a Laplace’s approximation32 to the total number of nuclei, we model the sharp spike in J(t) as Equations (1) and (2) might be solved using singular perturbation techniques32 because, as depicted in Figure 1, C changes over a much longer period than the duration of the nucleation burst which changes pT. However, J and n* are generally complicated and non-linear functions of C. Therefore, the analysis below focuses on general but approximate consequences of equations (1) and (2). Based on equations (1), (3), and (6), the concentration C for times near t = tm is

J ≈ J m exp  − 12 (t − t m ) 2 / σ II2 

i.e. ln[ J / J m ] = − 12 (t − t m ) 2 / σ II2 where higher order terms are omitted. To make equation (10) consistent with equation (9),

σ II2

must be defined as

1 C = Cm − 12 GJ m (t − tm ) 2 + L

σ II2

(7)

where higher order terms are not shown. Here Cm is the maximum concentration which is attained at t = tm, and Jm, to be determined, is the maximum nucleation rate which is

(10)

≡

GJ m * nm Cm

(11)

Now 2σ II can be interpreted as the approximate duration of the nucleation burst. See Figure S1 for a comparison of the classical nucleation theory rate expression with the



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corresponding Gaussian approximation to J(t) in equation (10). Based on equation (10) the total population of nuclei, ∞

pT∞ = ∫ dt J (t ) , is just a Gaussian integral 0

pT∞ ≈

2π Cm J m Gnm*

(12)

Because Jm would be difficult to measure directly, we attempt to eliminate it from equation (12). At time t = tm, Jm and pT (tm ) must satisfy Sugimoto’s balance equation:

0 = Q − J m nm* − G pT∞ / 2

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solution with a 10-fold excess of KCl solution as a reservoir of Cl- ions. The hydrolysis of diethyl sulfate (DES) in solution releases H+ ions which displace Ag+ ions from the gelatin into solution. The silver supply rate Q was controlled by changing the amount of DES that is added to the solution.28 The total number of nuclei was determined from TEM images of the final nuclei. Figure 2 shows

pT∞

vs. Q

as well as the values of G that are extracted from Sugimoto’s analysis and from our analysis. The best fit value of G can be used to confirm that ≪ 1 for all reasonable values of nm*, and therefore to confirm the validity of the equation pT∞ = 2Q / G for fitting.

(13)

Note that pT (tm ) = pT∞ / 2 according to equation (10). Combining (12) and (13) and solving for

pT∞ = =

pT∞

2Q  1 + 4a − 1    G  2a 

gives

(14)

2Q (1 − a + L) G

where

a≡

2Q nm* 2 ⋅ Cm G π

(15)

The factor Q/CmG within a is expected to be a very small number. It represents the ratio of the solute supply rate divided by the rate of solute consumption if all solutes at the maximum concentration were separately growing nuclei. Equation (14) is a general consequence of equations (1) and (2). It does not depend on any specific details of the nucleation rate expression. Thus (for sufficiently small a) our analysis confirms the most remarkable prediction from the work of Sugimoto et al.: the total number of nuclei is almost entirely controlled by the ratio Q/G, regardless of specific details in the nucleation rate expression. A few differences between our findings and those of Sugimoto should also be noted. First, our analysis predicts that large critical nuclei and slow growth kinetics can cause non-linear

pT∞

Figure 2: Inferring the growth rate G from the total population of nuclei and the monomer supply rate Q at the moment of the nucleation burst. The data points are from previous work by Sugimoto et al.28 PARTICLE SIZE DISTRIBUTIONS Sugimoto’s equation (4) predicts a sudden onset of nucleation with an exponential decaying rate. The corresponding particle size distribution (PSD) would increase exponentially up to a maximum size G(t-tm). Because equation (4) results in a qualitatively unrealistic PSD, Sugimoto separately used a population balance equation with a nucleation function of cosine shape to model PSDs.8 The cosine distribution yields a qualitatively realistic PSD, but it ispurely empirical. Our extended framework predicts the PSD directly from equations (1) and (2). The population density of nuclei of size n at time t’ is

∫ p (n, t ') =

t'

0

vs. Q relationships. The predicted

deviations from linearity may explain results from Hendricks et al. who observed a nonlinear

∞ T

p

vs. Q relation-

∞ T

ship. Second, our analysis predicts p = 2Q / G when a is small, while Sugimoto obtained pT∞ = Q / G . The additional factor of two influences the inferred (fitted) growth rate in the analysis of experimental data. To illustrate that the factor of two in the leading term of equation (14) doubles the inferred growth parameter G, we reanalyze the AgCl nucleation data of Sugimoto.28 Sugimoto prepared a gelatin with adsorbed Ag+ ions in

dtJ (t )δ [n(t ' | t ) − n]

∫

t'

0

(16)

dtJ (t )

Here δ is the Dirac delta, and n(t ' | t ) is the size of a nucleus at time t’ given that it was born at an earlier time t. The function n(t ' | t ) is

n(t ' | t ) = n *(t ) + (t '− t )G

(17)

where n*(t) is the critical nucleus size at time t. Particles born at each point in time have initial sizes and populations as indicated by n*(t) and by J(t)dt. After nucleation,


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they are transported along the diagonal lines with slope G (characteristics35) to the final time where a schematic of p(n) is shown. Figure 3 shows how these processes and the functions n*(t), J(t), and G determine the “final” particle size distribution p(n). For all times beyond the nucleation pulse, the PSD becomes a wave/bump that travels with a constant shape and constant velocity G. However, at times long after the burst the PSD will also be influenced by ripening and aggregation processes that are not included in our analysis.

the classical nucleation theory (CNT) as an illustrative example. According to CNT, J = Aexp[-B/ln2S] and n* = 2B / ln3 S with S=C/Csat and where A and B are weak functions of concentration, often taken to be constants. Cm, G, Q, and pT∞ = 2Q / G can be estimated from experimental data like that of Sugimoto et al., but direct estimates of Jm, nm*, and σII are more elusive. Even if we assume the nucleation mechanism is classical (it may not be), we cannot identify the A and B parameters without Jm, and one of the values nm* or σII. Fortunately, the PSD contains the required information to identify A and B. The total number of nuclei must satisfy equation (12) and analysis of the PSD with equation (19) provides an additional equation to identify both Jm and σII. A Gaussian approximation to J = Aexp[-B/ln2S] gives

 BGJ m  J ≈ J m exp  − (t − tm ) 2  3  Cm ln Sm 

Figure 3: As time increases toward tm, the critical nucleus size decreases (black curved arrow), the nucleation rate increases (yellow manifold over nucleus size curve), and earliest nuclei to form grow (along the characteristics35 in the (n,t) plane). The nucleation rate reaches a maximum at time tm and then begins to fall. The cumulative number of nuclei generated and carried along each characteristic. gives the PSD p(n). For times after the pulse, the PSD simply travels along the characteristics with no change in shape.

(20)

Then using Jm=Aexp[-B/ln2Sm] and 2σII2=Cmln3Sm/(BGJm) from the correspondence between equations (20) and (10) provides equations to find B and A. Figure 4(a) shows how different B-parameters lead to different PSD predictions. Figure 4(b) shows how different solute supply rates (Q) with the same B-parameter can be used to tune the PSD. (a)

The Supporting Information shows how the integral in equation (16) can be evaluated exactly. The exact PSD expression simplifies considerably under conditions where growth after nucleation is fast ( ≫ ∗/ ) and for times t’ after the nucleation burst ( t '− tm ≫ σ II ). Under these (b)

conditions the exact PSD approximately simplifies to

p (n, t ) ≈ J (t ( n)) / GpT∞

(18)

where t = t(n) is the time at which n*(t’) + G(t-t’) = n. If the critical nuclei are also very small compared to the nucleus sizes at the time of the PSD measurement, then equation (18) further simplifies to

p ( n, t ) ≈

 ( n − G (t − tm )) 2  Jm exp  −  ∞ GpT 2σ II 2G 2  

(19)

Equations (18) and (19) are important because they relate the PSD to the nucleation function. Given a theoretical nucleation rate expression, they also provide a way to extract its kinetic parameters from PSD data. Let us take

Figure 4: (a) The predicted effects of nucleation barrier parameter B on the PSD. (b) The predicted effects of (dimensionless) solute supply rate Q/GCsat on the PSD. All PSDs are shown at a time tm + 3σII.



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Faster solute supply rates lead to a narrower PSD because fewer nuclei are accumulated during the rapid approach to (and retreat from) the metastable zone limit. The reduced number of nuclei created en route to and from tm means fewer early and late nuclei which ultimately form the large and small tails of the distribution. Note that a faster solute supply rate also favors attainment of a slightly higher value of Sm, a subtle but evident aspect of Sugimoto’s AgCl and AgBr results. Larger B-parameters indicate a nucleation rate that is more sharply increasing with supersaturation. When the B-parameter increases, the nucleation rate more rapidly skyrockets upward from zero and thereby more sharply focuses the pulse of nucleation events. Finally note that the PSDs in Figure 4 were obtained by using illustrative B-parameters in the range 16 ≤ B ≤ 64 with G, Q, and Csat values from the AgCl experiments of Sugimoto et al. The resulting PSDs are millions of atoms wide, but by the time the particles reach 100nm in radius they contain ca. 100 million AgCl units. If G is truly sizeand concentration-independent, the distribution of sizes would still be ca. 1 million AgCl units wide, making the polydispersity extremely small. In reality, we expect the size- and concentration-dependence of growth as well as aggregation and ripening processes to alter conclusions about the long term size distributions. CONCLUSIONS Sugimoto et al. predicted that the number of nuclei generated ( pT∞ ) during a LaMer burst depends only on the solute supply rate (Q) and the growth rate (G). Specifically, Sugimoto et al. predicted that pT∞ = Q / G , independent of the nucleation kinetics. The prediction that Q controls the number of nuclei has been verified by some experiments, but the roles of growth and nucleation kinetics remain uncertain because of flaws in the original theoretical analysis. In this work, a more rigorous analysis of the coupled equations that govern of the concentrations of solutes and nuclei yields a modified prediction for pT∞ . Specifically, we find that pT∞ does depend on the nucleation kinetics, but that small nuclei and fast growth rates lead to the limiting expression pT∞ = 2Q / G , a result that confirms the limited role of the nucleation kinetics. The nucleation kinetics have a more pronounced effect on other aspects of the LaMer burst process. In particular, the nucleation kinetics determine the time and supersaturation at the nucleation burst. Moreover, we show that the particle size distributions (PSDs) directly reflect the nucleation kinetics. Given a PSD shortly after the nucleation burst, it should be possible to estimate the nucleation kinetics as a function of time during the nucleation stage. These results, in combination with experimental LaMer burst nucleation data, suggest new tests for theories of homogeneous nucleation kinetics. We note that certain assumptions in equations (1) and (2) facilitate analysis, but also limit the accuracy of our predictions. In particular, the growth rate G cannot actual-

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ly be size independent. In future work, we will develop deterministic models that include size- and concentrationdependent growth kinetics,26-27 and stochastic population balance models24-25 that self-consistently match the growth and nucleation fluxes as done in studies of other nucleation and growth phenomena.36-37 Finally, we note that PSD predictions at times long after the primary nucleation burst will require additional models of ripening38 and aggregation.39-40 ASSOCIATED CONTENT Supporting information The Supporting Information is available free of charge on the ACS Publications website at DOE: 10.1021/jacs.XXXXX Integration of equation (16) to obtain the PSD Accuracy of the Gaussian approximation to J(t) AUTHOR INFORMATION Corresponding author *[email protected] ACKNOWLEDGMENTS Peters was supported by the U.S. Department of Energy, Basic Energy Sciences, under grant DE-FG02-03ER15467. Chu was supported by a UC LEADS fellowship for undergraduate research at Santa Barbara. Owen was supported by a grant from the National Science Foundation NSF-CHE1710352. We thank Colin Unger, Susannah Scott, Linmin Wang, and Bryan Goldsmith for stimulating discussions. REFERENCES

1. Sear, R., Non-Self-Averaging Nucleation Rate Due to Quenched Disorder. J. Phys. Cond. Matt. 2011, 24, 052205. 2. Agarwal, V.; Peters, B., Solute Precipitate Nucleation: A Review of Theory and Simulation Advances. Adv. Chem. Phys. 2014, 155, 97-160. 3. LaMer, V. K., Nucleation in Phase Transitions. Ind. Eng. Chem. 1952, 44, 1270-77. 4. LaMer, V. K.; Dinegar, R. H., Theory, Production, and Mechanism of Formation of Monodispersed Hydrosols. J. Am. Chem. Soc. 1950, 72, 4847-54. 5. Gebauer, D.; Volkel, A.; Colfen, H., Stable Prenucleation Calcium Carbonate Clusters. Science 2008, 322, 1819-22. 6. Sun, Y., Controlled Synthesis of Colloidal Silver Nanoparticles in Organic Solutions: Empirical Rules for Nucleation Engineering. Chem. Soc. Rev. 2013, 42, 2497511. 7. Baronov, A.; Bufkin, K.; Shaw, D. W.; Johnson, B. L.; Patrick, D. L., A Simple Model of Burst Nucleation. Phys. Chem. Chem. Phys. 2015, 17, 20846-52. 8. Sugimoto, T., Monodispersed Particles. Elsevier: Amsterdam, 2001. 9. Kwon, S. G.; Hyeon, T., Colloidal Chemical Synthesis and Formation Kinetics of Uniformly Sized Nanocrystals of Metals, Oxides, and Chalcogenides. Acc. Chem. Res. 2008, 41, 1696-709.


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TOC Graphic


9


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TOC graphic 113x70mm (120 x 120 DPI)


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Figure 1 103x123mm (120 x 120 DPI)



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Figure 2 110x78mm (120 x 120 DPI)


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Figure 3 94x81mm (120 x 120 DPI)



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Figure 4 115x165mm (120 x 120 DPI)


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Nucleation and Growth Kinetics from LaMer Burst Data - The Journal

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