Article pubs.acs.org/cm
Gold Nanocluster Agglomeration Kinetic Studies: Evidence for Parallel Bimolecular Plus Autocatalytic Agglomeration Pathways as a Mechanism-Based Alternative to an Avrami-Based Analysis Eric. E. Finney, Shawn P. Shields, William E. Buhro, and Richard G. Finke* Department of Chemistry, Washington University, St. Louis, Missouri Department of Chemistry, Colorado State University, Ft. Collins, Colorado S Supporting Information *
ABSTRACT: The most extensive and highest quality Au0n nanocluster agglomeration size vs time TEM data set yet obtained are analyzed by a nanoparticle size vs time equation that is derived herein for parallel bimolecular (B + B → C, rate constant k3) and autocatalytic (B + C → 1.5C, rate constant k4) agglomeration steps of preformed nanoclusters, B. The results show that the size vs time data are well fit by the new size vs time equation. The fits and resultant k3 and k4 rate constants yield several interesting insights that are presented and discussed, including the finding that to date k4 > k3, that is, that the autocatalytic agglomeration rate constant is faster than the bimolecular rate constant, at least for the cases examined to date. The results of the effects of added TOABr (tetraoctylammonium bromide) on the 180 °C agglomeration k3 and k4 rate constants in unstirred diphenylmethane solvent are also presented and discussed, the TOABr being added originally to compact the nanoclusters double layer thereby helping induce agglomeration. The observed different [TOABr] effects on k3 vs k4 also provide prima facie evidence that the two agglomeration steps are fundamentally different and unique. Literature size vs time data, from El-Sayed et al. for Pd nanocluster agglomeration, are also fit as a further test of the new, mechanism-based size vs time equation. The combined results, showing good fits by the k3 and k4 steps to the Au0n as well as literature Pd, Pt, and Ir nanocluster data, provide good support for the underlying B + B → C and B + C → 1.5C agglomeration steps themselves as well as for the assumptions and math behind the new size vs time equation. The significance of the results in general, as well as for future measurements of k3 and k4 rate constants as a preferred way to quantitate nanocluster stability in solution, are also presented and discussed. Most significant, however, is that as a result of the present work one can now use chemical equations and associated, mechanistically rigorously defined concepts of bimolecular (B + B → C; rate constant k3) and autocatalytic (B+ C → 1.5C; rate constant k4) agglomeration to analyze and describe nanoparticle agglomeration rather than the harder to interpret, more obscure n and k parameters from an Avrami-type, semiempirical curve fit. KEYWORDS: bimolecular agglomeration, autocatalytic agglomeration, 2- and 4-step mechanisms of nanocluster nucleation, autocatalytic growth and agglomeration, mechanism-based nanocluster size vs time equation, mechanism-based substitute for Avrami equation analyses
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INTRODUCTION Understanding the mechanisms of nanocluster formation, including any agglomeration1,2 that may be occurring, is important because this can lead to greater control over nanocluster size and, therefore, properties. Because of this, prior work in our respective laboratories at Washington University and Colorado State University has been devoted to elucidating the mechanisms of nanocluster nucleation, growth, and agglomeration (vide infra). Throughout this paper we will use the term agglomeration1 (to “collect or form into a mass or group”), rather than other possible terms,1 to describe the reactions studied herein. At Washington University, previous work demonstrated that heterogeneous seeded growth of nanoclusters (i.e., using seeds of a different element) could lead to enhanced kinetic control, and therefore narrow size distributions.3 Small Au clusters were used to grow nanoclusters of Bi, In, and Sn with size dispersions between 5 and 13%. Such near-monodisperse In nanoclusters © 2012 American Chemical Society
have been used in the preparation of InP quantum wires with diameters that could be controlled by the size of In cluster used.4 Bi nanoclusters formed via homogeneous nucleation and growth, via the decomposition of Bi[N(SiMe3)2]3 in the presence of Na[N(SiMe3)2],5 are currently used in the preparation of nanowires.6 More recent work at Washington University has focused on the important role of agglomeration in the growth of nanoclusters, including work on thiolate-capped Au nanoparticles, Scheme 1,7 Ag nanoparticles,8 and the aforementioned Bi nanoparticle system.9 The Washington University work provides evidence that agglomeration is a more important part of nanoparticle formation than is commonly recognized and also provides clear examples of Ostwald ripening as well as Received: October 25, 2011 Revised: March 30, 2012 Published: May 7, 2012 1718
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Scheme 1. Generalized Reaction Scheme for (I) the Formation of the Au0 Nanoclusters from the Reduction of HAuCl4·3H2O in the Presence of Decanethiol (DT) and Tetraoctylammonium Bromide (TOABr), Followed by (II) Their Thermally Induced Agglomeration in the Presence of Varying Amounts of TOABra,19)
a The precise surface ligand composition of the Au0n nanoclusters is not known (as is common at present19), so that possible additional stabilizing ligands, and their amounts m-r, are written below to remind us of this compositional uncertainty (and because the resultant stabilization and agglomeration tendency are expected to reflect the surface ligands that are present).
first place, Scheme 1? These are some of the key problems that, ultimately, are addressed in the following, collaborative work. The interested reader is directed to a 2009 publication for further discussion and examples of these important points and the current state of confusion in Avrami-based treatments of solid-state kinetics10not to mention the confusion caused by using a solid-state model for solution kinetic processes such as agglomeration. Indeed, it was only since the fits reported herein were done before the prior publication,7 fits to chemically welldefined equations with their thereby rigorously defined rate constants, that we were able to coin and use the more sensible terms “aggregative nucleation” and “aggregative growth” in that prior publication7 (terms which, on the basis of the present paper, are however probably better replaced by the terms “agglomerative nucleation” and “agglomerative growth”, vide infra). Such precise definitions, concepts and nomenclature matter greatly in science, of course, since it is ultimately the precision (or lack thereof) in those concepts and their associated language that expresses our understanding (or lack thereof) in the underlying, basic science. In work at Colorado State University, studies have focused on the nucleation and growth of transition-metal nanoclusters in solution, leading to a minimalistic 2-step mechanism for transition-metal nanocluster nucleation and growth, the so-called Finke−Watzky (hereafter F−W) mechanism,11 the first two steps in Scheme 2. In Scheme 2, A is a nanocluster precursor complex (commonly a metal salt, e.g., A = AuIII in Scheme 1), and B is the nanocluster product (B = Au0∼146 in Scheme 1, for example), the precise definitions of A and B being set by the reaction stoichiometry, A → B (or, again using Scheme 1 as an example, AuIII → Au0). More recently, two more steps were added to the 2-step, Colorado State mechanism to account for nanocluster (B)
criteria for detecting when Ostwald ripening is,8,9 or is not,7 present. Initially, and following the prior literature cited elsewhere,7 an Avrami-type (KJMA)7 model was used to fit the sigmoidal nanocluster mean volume, V̅ t, divided by limiting final volume, V̅ lim, (i.e., V̅ t/V̅ lim) vs time agglomeration kinetic data, resulting in Avrami exponent, n, and Avrami rate parameter, k, values. This in turn yielded the well-known problems of: (i) how to interpret the Avrami n value, n historically being interpreted as the “dimensionality of the nucleation”, so that the physical meaning of the “dimension of a solution nanoparticle agglomeration (i.e., of nucleative agglomeration)” became one issue. In addition, the meaningful interpretation of the n value was compounded by the fact that the fits are insensitive to constrained values of n = 1−3 (“dimensions”, n, 3−4 being reported in the Avrami literature as discussed and referenced elsewhere10). The Avrami equation fits also yielded the problem of (ii) the precise physical meaning of the Avrami rate parameter, knote in this regard, and as discussed elsewhere,10 the Avrami k is not a traditional, rigorous rate constant defined in the normal way by a specific, balanced chemical reaction step. Moreover, there was a third, significant, more general problem of (iii) what precise, proper and well-defined wordsthat is, what physical and chemical conceptsto use to describe the agglomeration process? Should it be “nucleation and growth” (i.e., but for the agglomeration process) for example, as was common in the prior literature summarized elsewhere?7 If so, how can one talk of two chemical processes without any rate constantsand only one rate “parameter”? If one uses “nucleation and growth” (of agglomeration), then how are those terms to be distinguished from the (true) initial nucleation and growth of the AuIII precursor to yield the ca. 1.68 nm Au0 nanoparticles in the 1719
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and autocatalytic growth is the underlying mechanism of nanoparticle formation (i.e., the first two steps in Scheme 2).15 The starting point for that size vs time equation is eq 1 below for the diameter (D) of a cluster; in eq 1 N is the number of atoms in the nanocluster, MW is the molecular weight of the metal comprising the cluster, ρ is the bulk density of the metal, and NA is Avogadro’s number. Two assumptions15 behind eq 1 are that the nanoclusters are spherical in shape (as they are for the studies herein) and that the bulk density can be used to approximate the nanocluster density. The latter assumption likely leads to a slight overestimate of the diameter of the clusters because the true nanocluster density is likely a bit higher because of the slightly compacted nature expected and seen for transition-metal nanoclusters.15
Scheme 2
⎛ 6NMW ⎞1/3 D=⎜ ⎟ ⎝ πρNA ⎠
bimolecular agglomeration to larger metal particles C,12 B + B → C, and then also a novel autocatalytic agglomeration step, B + C → 1.5C (where the 1.5 stoichiometry factor is required since there are 2 B in the initial C that is formed).13 In terms of the Au0 system in Scheme 1, C is the larger, agglomerated Au0n nanoclusters, that is, C = Au0n. The full, 4-step mechanism is shown in Scheme 2 and consists of precisely defined nucleation, autocatalytic growth, bimolecular agglomeration, and autocatalytic agglomeration steps with their associated rate constants k1, k2, k3, and k4 defined by those balanced equations. Independent evidence has recently been published for the last two steps of bimolecular plus autocatalytic agglomeration in Scheme 2 in the case of Ir0n nanoclusters.14 A total of more than 21 mechanisms have been ruled out13 in reaching the minimalistic, “Ockham’s razor” mechanism shown in Scheme 2. It is important to note that, since any nanocluster nucleation and growth to a cluster containing ca. 146 atoms, such as an averaged sized, ∼1.68 nm Au0∼146 nanocluster, must have at least 146 steps in the mechanism, the 4 equations below are sums of many reactionsthat is, pseudoelementary steps, the pseudoelementary concept being a key concept for dealing with the kinetics of complex, multistep chemical reactions. The reader interested in understanding better the pseudoelementary step concept and the other details here is refereed to a series of 10 papers on the kinetics and mechanism of nanoparticle nucleation and growth that date back to 1997.10−18 Because agglomeration is an important process in nanocluster formation, and because a cluster’s size can affect its properties, the agglomeration of colloids or nanoclusters has been the subject of numerous prior studies.2,7−9,12,13 In the area of transition-metal nanoclusters, recent work at Colorado State University has addressed agglomeration, focusing on the last two steps of Scheme 2 by studying the agglomeration of preformed nanoclusters. For example, evidence has been presented suggesting that the first agglomeration step, B + B → C, proceeds via the indicated bimolecular combination of nanoclusters, B (and with activation parameters consistent with an associative mechanism), while the autocatalytic agglomeration step, B + C → 1.5C, is also bimolecular but exhibits activation parameters that imply an underlying dissociative process (e.g., one in which the ligand or possibly solvent is dissociated from the cluster’s surface before agglomeration can take place), at least for the Ir nanoparticle system examined.14 More recently, work at Colorado State University has led to the derivation of an equation expressing nanocluster size vs the reaction time, the k1 and k2 rate constants, and the initial concentration of A, [A]0 (see the first two steps in Scheme 2)11 when the F−W 2-step mechanism for nanocluster nucleation
(1)
In the present report, we are focusing on the agglomeration of existing nanoclusters, so it is the last two, agglomeration steps in Scheme 2 that are relevant to the present work where we follow the growth of the agglomerates, C (vs as our previous study of nanocluster size vs time which followed the growth of the parent nanoclusters,15 B). The number of atoms in a cluster at a given time, Nt, can be related to the concentration of clusters in solution via eq 2. Nt ∝
[C ]t VsolnNA number of nanoclusters
(2)
Although the number of clusters in the solution cannot be accurately calculated without knowing the final nanocluster size, knowledge of the final size, Df, allows us to use eq 3 to relate the diameter of the cluster C at any given time, [C]t, to the final diameter and the concentration of the smaller nanocluster B at that time. The factor of 2 in eq 3 is required because of the 2B:1C stoichiometry of the third step shown back in Scheme 2. A detailed derivation of the size vs time eq 3 is provided in the Supporting Information for the interested reader (see also the derivation elsewhere15 if additional details are desired). The important part for the present study is that, using eq 3 and the kinetics of agglomeration (vide infra), we can analyze nanoparticle size vs time data to obtain the rate constants k3 and k4 for the agglomeration process. ⎛ 2[C]t ⎞1/3 Dt ∝ Df ⎜ ⎟ ⎝ [B]0 ⎠
(3)
Herein we analyze data obtained at Washington University for the agglomeration of preformed Au nanoclusters, stabilized by decanethiol (DT),19 in the presence of different concentrations of tetraoctylammonium bromide (TOABr), Scheme 1, vide supra. We show (i) that the agglomeration of Au nanoclusters can be fit with just the last two, agglomeration steps of the mechanism in Scheme 2, (ii) that a size vs time equation can be derived for this 2-step agglomeration mechanism, and then (iii) that the resultant diameter vs time equation can be used to fit TEM size vs time data for the Au nanocluster agglomeration. This allows us to determine the values of the agglomeration rate constants k3 and k4, thereby kinetically quantifying the agglomeration kinetics of this Au nanocluster system, the first such measurements for Au0 nanoclusters, only the second such measurement of such agglomeration kinetics14 using the 4-step mechanism, and the first case where such extensive TEM 1720
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the fits shown in the Supporting Information (Figure S1) demonstrate that the data are fit reasonably well using the first two nucleation and growth steps of Scheme 2 rather than the (more appropriate, vide infra) two agglomeration steps. However, the Au0∼146 nanoclusters employed as starting materials have already been formed, that is, have already been nucleated and grown, so that we know are starting with B (nanoclusters) and not A (the HAuIIICl4·3H2O nanocluster precursor) in this system. In other words, because we are by definition studying the agglomeration of preformed nanoclusters, logically we should use the agglomeration steps of Scheme 1 and not the nucleation steps for fitting and discussion of the data. The fact that the nucleation and growth steps could have been used to fit the data, when the agglomeration steps are what need to be used to be conceptually correct, points to the critical importance of definitions and knowing one’s starting pointand the confusion that would otherwise quite easily be encountered if one just uses the words “nucleation and growth” without those words and starting point being defined by balanced, chemical reactions and mechanistic equations. Some additional discussion of the source of, and the mechanistic justification for, the terms used previously7−9 of “aggregative nucleation” (i.e., nucleation of aggregation) and “aggregative growth” (i.e., (autocatalytic) growth by aggregation) is provided in the Supporting Information for the interested reader. As discussed in that Supporting Information, alternative terms with a more rigorous mechanistic basis when the 4-step mechanism applies kinetically are, respectively, “bimolecular agglomeration” (or equivalently “bimolecular aggregation”) and “autocatalytic agglomeration” (or equivalently “autocatalytic aggregation”). As a second alternative mechanistic hypothesis, we also tried fitting the data using only the B + B → C, k3 step in Scheme 2, without the autocatalytic agglomeration step. The resulting, inferior fits are given in the Supporting Information, Figure S2. Assumptions behind the Use of Eq 5 for Fitting Cluster Size Vs Time Data. Above we provided two of the assumptions underlying the use of the nanocluster size vs time equation, namely that the clusters are spherical and that we can use the bulk density as a first approximation for the nanocluster’s probably slightly higher density. Another underlying assumption is that the starting nanoclusters, B, are all of one average size to start (i.e., essentially monodisperse to start) so that eq 5 provides the average size of C vs time. Note that eq 5 cannot provide cluster size distributions.15 In addition, we necessarily assume that [C]0 = 0; that is, we assume that the preformed nanoclusters have not yet aggregated and are just B to start. One result of this assumption is that the initial size of the larger, agglomerated clusters must be 0, because we are specifying C as the cluster whose growth we are following. Additional assumptions behind the general treatment that the derivation of eq 5 follows are given elsewhere15 for the interested reader.
diameter vs time data are both available and have been employed. Significantly, we then return to the questions raised in the Introduction and are able to then use proper chemicalequation-defined words and concepts to describe the two agglomeration processes that are occurring.
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EXPERIMENTAL SECTION
The experimental procedures for the preparation and agglomeration of the Au nanoclusters in the presence of varying amounts of tetrabutylammonium bromide (TOABr) are detailed fully in a preceding paper.7 Fitting the Size Vs Time Data to the Bimolecular Plus Autocatalytic Agglomeration Mechanism, B + B → C, B + C → 1.5C. Because the experiments herein are following the agglomeration of preformed nanoclusters, by definition B, the last two steps of the 4-step, double autocatalytic mechanism, Scheme 2 (i.e., the steps involving agglomeration), were used to fit the size vs time data. The integrated rate equation for the two agglomeration steps is shown in eq 4, presented herein for the first time (see the Supporting Information for the derivation of eq 4).
[C]t =
2k 3[B]0 (1 − e k4[B]0 t /2) 4k 3(1 − e k4[B]0 t /2) − k4
(4)
Combining this expression for [C]t with the version of eq 3 in the Supporting Information that expresses eq 3 as a full equality yields eq 5 (for the detailed derivation of eq 5, please see the Supporting Information). The resultant eq 5 gives the size of a cluster at any time given the initial nanocluster size, the final size of the agglomerates, the initial concentration of nanoclusters, and the agglomeration rate constants k3 and k4.
⎡ ⎤1/3 4k 3(1 − e k4[B]0 t /2) 3 3 ⎥ Dt = ⎢D0 3 + ( D − D ) f 0 ⎢⎣ 4k 3(1 − e k4[B]0 t /2) − k4 ⎦⎥
(5)
The curve fitting was accomplished using Microcal Origin ver. 7.0. The uncertainties in k3 and k4 were provided by the Origin program, as were R2 values to determine how closely the equation fit the data. The errors in the size data shown in the figures in the main text were calculated as the standard deviations in the TEM-measured sizes of the clusters (for further details see the Supporting Information elsewhere7). The value of [B]0 in this system was estimated based on the average diameter of the nanoclusters at the beginning of the agglomeration experiments, 1.68 nm, and the initial amount of HAuCl4·3H2O used to make the nanoclusters, 0.55 mmol. Specifically, starting with eq 1 and solving for N gives N = 146 atoms per starting Au0 nanocluster on average, Au0∼146. Starting with 0.55 mmol of Au atoms (3.3121 × 1020 total Au atoms), one can calculate the number of moles of clusters, eq 6:
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RESULTS AND DISCUSSION The size vs time curves for nanocluster agglomeration in the presence of varying amounts of TOABr are given in Figure 1, along with the fits to eq 5 for each data set. As mentioned in the Experimental Section, the error bars shown for the diameters of each datum point in Figure 1, and the other figures which follow, were taken as the standard deviations from the TEM size measurements. Several results are of immediate interest: (a) first, the agglomeration is well accounted for (i.e., the curves are well fit, within the experimental error of the data) by the 2-step agglomeration mechanism (attempts to fit the size data using just the first, B+B→C, step failed; see the Supporting Information, Figure S2). The good fits to the 2-step agglomeration mechanism provide support for that mechanism. Overall, the results provide the most extensive and best nanocluster size vs time data, and thus the most stringent test to date using size vs time data, of the two agglomeration steps that are part of the overall 4-step mechanism of nanocluster
3.3121 × 1020clusters 146 atoms/cluster = 2.27 × 1018 clusters = 3.77 × 10−6 moles of clusters
(6)
Because half of the solution was used for each study, and a total solution of 5.7 mL was made,7 the value of [B]0 is calculated in eq 7 1 2
× 3.77 × 10−6 moles 0.0057 L
= 0.000330 M
(7)
−4
Therefore, a value of [B]0 = 3.30 × 10 M is used for the experiments herein. As an alternative hypothesis for analysis of the size vs time kinetic data, we also tried to fit the data using the first two steps in Scheme 1 (i.e., the A → B, A + B → 2B nucleation and growth steps). Our expectation was that use of the formally nucleation and growth steps would also fit the data, since the nucleation-and-growth and the agglomeration mechanisms both have an autocatalytic step. Indeed, 1721
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Figure 1. Size vs time curves for the agglomeration of decanethiol-stabilized Au0 clusters in the presence of varying amounts of tetraoctylammonium bromide (TOABr) in unstirred diphenylmethane at 180 °C. The concentration of TOABr is given within each curve. Each curve is fit to the B + B → C, B + C → 1.5C bimolecular plus autocatalytic agglomeration mechanism; the associated rate constants k3 and k4 are given within each curve, as well as the R2 values to judge the quality of the fits. The fit to the [TOABr] = 0.264 M data is statistically less good (R2 = 0.9257), but is improved if the point at t = 105 min is excluded as an apparent outlier (R2 = 0.9513, Supporting Information, Figure S3).
stabilized by Cl− anions13 and to Ir nanoclusters stabilized by polyoxoanions.14 Fourth and additionally, (d) the resultant rate constants k3 and k4 are of interest, specifically the result that the autocatalytic rate constant k4 is larger than k3 (k4 being 1500 times faster than k3 for the case of 0.264 M TOABr, to only 15-fold
formation and agglomeration13,14 back in Scheme 2. Second, (b) the good fits seen provide ex post facto, a posteriori support for the derivations and data treatment given herein; and (c) third, the data provide important evidence for the greater generality of the two agglomeration steps reproduced in Scheme 213,14 because it previously had been applied only to Pt nanoclusters 1722
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faster when 0.681 M TOABr is present). It is of interest that the same k4 ≫ k3 trend is seen for Cl− stabilized Ptn as well as for polyoxoanion-stabilized Irn nanoclusters,13,14 so that decanethiol-stabilized Aun nanoclusters now join and reinforce this same trend. Although the precise physical reason behind the observation that k4 ≫ k3 is presently not known for certain, one good guess is that k4 ≫ k3 is related to the decreasing metal−ligand bond energy with increasing size nanoclusters,13 thereby allowing faster agglomeration in the large, C, aggregates (the added implication being that nanocluster surface metal− ligand cleavages are part of the agglomeration pathway, as one might expect). Fortifying the above explanation for k4 ≫ k3 are the three alternative hypotheses for this observation that we considered but can rule out, hypotheses discussed in the Supporting Information for the interested reader. Noteworthy here is that the k4 pathway is intrinsically size-focusing13c (smaller B plus larger C go to a more uniform size), consistent with the observation of a change in size from 1.68 nm ±0.35 (i.e., ± 21%) to larger, less disperse 5.37 ± 0.38 (i.e., ± 7.1%) agglomerates. Overall, the results provide additional evidence consistent with and supportive of the existence of the novel autocatalytic agglomeration pathway,13,14 as well as for the kinetic importance of that k4, autocatalytic agglomeration pathway. Another interesting set of results is how the rate constants k3 and k4 depend on the amount of TOABr in the solution. TOABr was added since added salts are expected to compact the double (really multi-) layer of anion-repulsion-stabilized (i.e., DLVO-theory19 stabilized) nanoparticles,19 thereby accelerating their agglomeration. Very interestingly, the bimolecular agglomeration rate constant k3 begins to increase past ca. 0.45 M TOABr, Figure 2. The autocatalytic agglomeration rate
Figure 3. Dependence of the autocatalytic agglomeration rate constant k4 on the concentration of TOABr for the agglomeration of decanethiol-stabilized Au0 clusters in diphenylmethane at 180 °C. In composite, the data would appear to peak somewhere around the ca. 0.26 to ca. 0.45 M TOAB range.
Specifically, the (apparently associatively activated14) k3 step is made faster at higher TOABr where the increased ionic strength collapses the nanocluster’s double (really multi-) layer,19 thereby increasing agglomeration and k3. On the other hand, the (apparently dissociatively activated14) k4 step appears to be slowed by higher TOABr concentrations, the rationalization here being that the added Br− serves as a nanocluster ligand, poisoning open coordination sites on the nanocluster that are required for k4-based agglomeration. Regardless of whether these rationalizations of the observed TOABr effects are precisely correct, the different k3 vs k4 responses to added TOABr provide additional evidence13 for the chemically distinct nature of the k3 and k4 agglomeration processes. The overall, final cluster size shows a small, apparent dependence on the concentration of TOABr, Figure 4, as has
Figure 2. Dependence of the bimolecular agglomeration rate constant k3 on the concentration of TOABr for the agglomeration of decanethiol-stabilized Au0 clusters in diphenylmethane at 180 °C. Figure 4. Dependence of the final nanocluster size on the concentration of TOABr for the agglomeration of decanethiolstabilized Au0 clusters in diphenylmethane at 180 °C.
constant, k4, on the other hand does not show this trend of increasing past ca. 0.45 M TOABr and even appears to pass through a maximum near [TOABr] = 0.45 M, Figure 3 (the kg parameter and plot vs [TOABr] in Figure 8 elsewhere7 also suggests that there is a maximum, although the large error in k4 for the [TOABr] = 0.264 M data point makes this conclusion tentative at present). What is of further interest here is that these observations, the first of their kind, appear to be consistent with and supportive of the previously suggested associative, vs dissociative, natures of the k3 and k4 steps, respectively.14
been previously noted and discussed.7 That increase in size is (at most) 1.7 nm over the range of [TOABr] studied, although the alternative, more rigorous conclusion here is that all the nanoclusters in Figure 4 are the same size within 5.7 ± 0.7 nm (1σ error estimates). Other, Literature Agglomeration Systems. In a search for other metal nanocluster agglomeration systems that start 1723
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(i) The agglomeration of TOABr and decanethiol/thiolatestabilized Aun nanoclusters at 180 °C can be quantitatively accounted for by the bimolecular plus autocatalytic agglomeration kinetic scheme,13,14 B + B → C, B + C → 1.5C; (ii) The good fits observed to the data provide support for the equations derived and justification of the assumptions employed in their derivation and use; (iii) The results provide the best experimental test to date, in terms of diameter vs time data, of the parallel, B + B → C and B + C → 1.5C agglomeration pathways; the results also provide further support13 for the novel autocatalytic agglomeration pathway, B + C → 1.5C as well as its tendency to dominate kinetically; (iv) The resulting rate constants are of interest and show, as before for Ir and Pt nanoclusters, that k4 > k3. These results, as well as the added TOABr results, are in general consistent with the finding of a weaker metal−ligand bond energy in the larger, agglomerated nanoclusters, C,13,14 although verification or refutation of the hypothesis of size-dependent nanocluster metal−ligand bond energies remains to be accomplished. (v) The kinetic fits confirm in a chemically precise way the previous conclusion7 that Ostwald ripening is not present in this Au nanoparticle agglomeration system (i.e., and since the equations employed herein do not contain any Ostwald ripening, yet yield good fits to the data); (vi) The chemical equations and associated, mechanistically rigorously defined concepts of bimolecular (B + B → C; rate constant k3) and autocatalytic (B + C → 1.5C; rate constant k4) agglomeration can be used to describe the agglomeration reactions that are occurring. Their use yields precisely defined, readily interpretable (i.e., “normal”) rate constants, k3 and k4, that can be used as replacements10 for the harder to interpret, more obscure n and k parameters from an Avrami-type semiempirical curve fit, parameters that we have shown10 are complex convolutions of k3 and k4 in any event. Overall, these studies plus our 2008 paper14 provide the necessary background studies for more extensive investigations of nanocluster agglomeration en route to better control and thus exploitation when desired, or avoidance when desired, of agglomeration in nanocluster syntheses. These and or earlier results14 also set the stage to measure and then exploit agglomeration k3 and k4 rate constants as a preferred way to quantitate nanocluster stability.14,19 As such, it is our hope that the present contribution will prove valuable to the community in future analyses of nanoparticle agglomeration kinetic and other data.
with preformed nanoclusters, we came upon data from ElSayed et al.’s study of the Pd nanocluster-catalyzed Suzuki coupling between phenylboronic acid and iodobenzene.20 As is shown in the Supporting Information, the agglomeration of Pd nanoclusters during the course of this reaction can also be well fit by the B + B → C, B + C → 1.5C mechanism. Those Pd agglomeration results, along with the Pt,13 Ir,14 and present Au nanocluster agglomerations, begin to provide a strong case for the greater generality of the B + B → C, B + C → 1.5C agglomeration mechanism,13 a mechanism obtained after ruling out initially 18 (and now 21)13 total other conceivable mechanisms en route to the overall 4-step shown back in Scheme 2. Primary New Results from This Study. There are, then, 11 main new results due to the present work, specifically: (i) the [C]t vs time eq 4 and then associated specific eq 5; (ii) the treatment of Au0 agglomeration kinetic data by the last 2 steps of the 4-step mechanism (all prior data being for Ir or Pt nanoclusters); (iii) the treatment of the best available, most extensive size vs time (Au0) nanocluster agglomeration data herein by these new equations; (iv) the resultant k3 and k4 rate constants for the Au0 systemnever before available agglomertion rate constants for the Au0 system; (v) the insight that the [TOABr] effects on k3 and k4 are different, offering independent evidence for the suggestion and discussion elsewhere13 that these agglomeration steps are fundamentally different as their stoichiometries and activation parameters indicate; (vi) the treatment herein of the literature, Pd0 agglomeration data; (vii) evidence that other mechanism tried of just bimolecular agglomeration alone, B + B → C, cannot adequately fit the data, that is, disproof of this alterative mechanism for the present Au0 agglomeration example, such disproofs being at the heart of rigorous mechanistic science; and (viii) the first compelling size-vs time evidence, along with the kinetic fits, confirming that autocatalytic agglomeration, B + C → 1.5 C, is size focusing as had been postulated. Lastly, but probably most significantly, the present paper (viii) provides an alternative treatment forthat is, an alternative hypothesis tothe Avrami-based analysis more commonly used in both the general agglomeration literature as well as in the prior paper,7 such attempted disproof of multiple alternative hypotheses being the essence of reliable, rigorous science.21 Hence, a very important part of the present work is (ix) that it offers balanced-chemical-reaction-based equations and hence concepts, words, and rigorously defined rate constants for the Au0 agglomeration example for the first time, namely bimolecular agglomeration, B + B → C (rate constant k3) and size focusing autocatalytic agglomeration, B + C → 1.5 C (rate constant k4). Also useful and new is (x) the treatment showing the first 2, nucleation and growth, steps of Scheme 2 can also fit the data, but that using those descriptors can only result in confusion since the Au0 nanoparticles were already “nucleated and grown” from the AuIII starting material in Scheme 1. Hence, also important is (xi) the contribution of the correct concepts and words of “aggregative (or better agglomerative) nucleation” and “aggregative (or better agglomerative) growth” to both the prior paper,7 and the literature in general, as is the simple but important contribution of the preferred concepts and words of bimolecular agglomeration and autocatalytic agglomeration, at least from a chemical and mechanistic view point. In short, the above 11 new contributions have resulted from the present work.
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ASSOCIATED CONTENT
S Supporting Information *
Some additional discussion of the source of, and the mechanistic justification for, the terms used previously of “aggregative nucleation” (i.e., nucleation of aggregation) and “aggregative growth” (i.e., (autocatalytic) growth by aggregation); alternative hypotheses for the k4 > k3 observation that were considered; derivations of the integrated rate equation and the cluster size vs time equation for the B + B → C, B + C → 1.5C agglomeration mechanism; attempted fits of the data to the F-W 2-step, nucleation and growth mechanism; attempted fits of the data to just the B + B → C, bimolecular agglomeration step; fit of the 0.235 M TOABr data with the outlier excluded; and fitting of El-Sayed’s data for Pd nanocluster agglomeration during a Pd nanocluster-catalyzed Suzuki coupling reaction.This material is available free of charge via the Internet at http://pubs.acs.org.
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CONCLUSIONS The main conclusions from the present report and its new results, then, are that: 1724
dx.doi.org/10.1021/cm203186y | Chem. Mater. 2012, 24, 1718−1725
Chemistry of Materials
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Article
References 1−8 therein contain a detailed listing of the agglomeration literature up to 2004.) (13) (a) Besson, C.; Finney, E. E.; Finke, R. G. J. Am .Chem. Soc. 2005, 127, 8179−8184. (b) Besson, C.; Finney, E. E.; Finke, R. G. Chem. Mater. 2005, 17, 4925−4938. (c) Finney, E. E.; Finke, R. G. Chem. Mater. 2008, 20, 1956−1970. (14) Ott, L. S.; Finke, R. G. Chem. Mater. 2008, 20, 2592−2601. (15) Watzky, M. A.; Finney, E. E.; Finke, R. G. J. Am. Chem. Soc. 2008, 130, 11959−11969. (16) Starkey-Ott, L.; Finke, R. G. J. Nanosci. Nanotechnol. 2008, 8, 1551−1556. (17) Finney, E. E.; Finke, R. G. Chem. Mater. 2009, 21, 4468−4479. (18) Finney, E. E.; Finke, R. G. J. Colloid Interface Sci. 2008, 317, 351−374 and references therein. (19) Ott, L. S.; Finke, R. G. Coord. Chem. Rev. 2007, 251, 1075− 1100. (20) (a) Narayanan, R.; Tabor, C.; El-Sayed, M. A. Top. Catal. 2008, 48, 60−74. (b) Narayanan, R.; El-Sayed, M. A. J. Am. Chem. Soc. 2003, 125, 8340−8347. (21) Platt, J. R. Science 1964, 146, 347.
AUTHOR INFORMATION
Corresponding Author
*E-mail: rfi
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Dr. Murielle Watzky, as well as Aimee Morris and Joe Mondloch, for their help in checking the equations and derivations herein as well as other critical commentary that improved the final presentation. The collection of the data utilized herein at Washington University was accomplished via NSF Grant CHE-1012898. The work at Colorado State University was accomplished via DOE support from the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, grant number DE-FFD02-03ER15453.
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REFERENCES
(1) The definition of agglomeration cited is that found in Google’s dictionary (other dictionaries examined had similar definitions), agglomeration typically being defined as either (i) “a whole formed by combining several (typically disparate) elements”, or (ii) “a material or structure formed from a loosely compacted mass of fragments or particles”; one should be aware, however, that agglomeration is also called “aggregation”, “coagulation”, or “flocculation” in the literature, as briefly discussed elsewhere.12 (2) Some lead references to agglomeration; additional references are provided in our earlier papers in this area:7−10 (a) May, M. M.; Zheng, W.; Leibowitz, F. L.; Ly, N. K.; Zhong, C.-J. Langmuir 2000, 16, 490− 497. (b) Van Hyning, D. L.; Klemperer, W. G.; Zukoski, C. F. Langmuir 2001, 17, 3128−3135. (c) Penn, R. L.; Banfield, J. F. Science 1998, 281, 969−971. (d) Peng, X.; Wickham, J.; Alivisatos, A. P. J. Am. Chem. Soc. 1998, 120, 5343−5344. (e) Bogush, G. H.; Zukoski, C. F. J. Colloid Interface Sci. 1991, 142, 19−34. (f) Meakin, P. Adv. Colloid Interface Sci. 1988, 28, 249−331. (g) Weitz, D. A.; Huang, J. S.; Lin, M. Y.; Sung, J. Phys. Rev. Lett. 1985, 54, 1416−1419. (3) Yu, H.; Gibbons, P. C.; Kelton, K. F.; Buhro, W. E. J. Am. Chem. Soc. 2001, 123, 9198−9199. (4) Yu, H.; Li, J.; Loomis, R. A.; Wang, L.-W.; Buhro, W. E. Nat. Mater. 2003, 2, 517−520. (5) Yu, H.; Li, J.; Loomis, R. A.; Gibbons, P. C.; Wang, L.-W.; Buhro, W. E. J. Am. Chem. Soc. 2003, 125, 16168−16169. (6) Wang, F.; Dong, A.; Sun, J.; Tang, R.; Yu, H.; Buhro, W. E. Inorg. Chem. 2006, 45, 7511−7521. (7) Shields, S.; Richards, V. N.; Buhro, W. E. Chem. Mater. 2010, 22, 3212−3225. See also the references therein (e.g., as detailed on p. 3213) to prior studies of agglomerative processes in nanoparticle growth. (8) Richards, V. N.; Rath, N. P.; Buhro, W. E. Chem. Mater. 2010, 22, 3556−3567. See also the references therein (e.g., as detailed on p. 3566) to prior studies of agglomerative processes in nanoparticle growth. (9) Richards, V. N; Shields, S.; Buhro, W. E. Chem. Mater. 2011, 23, 137−144. (10) Finney, E. E.; Finke, R. G. Chem. Mater. 2009, 21, 4692−4705. The correspondence shown therein is between the n and k of Avramitype equations and k1 and k2 (i.e., the first 2 steps of the 4-step mechanism). However, since [B] is effectively constant during the induction period, k3[B] = k3,obs so that the k3 and k4 steps of the 4-step mechanism can be substituted for, with mathematical equivalence to, the k1 and k2 steps and rate constants used in the above cited paper. (11) Watzky, M. A.; Finke, R. G. J. Am. Chem. Soc. 1997, 119, 10382−10400. (12) Hornstein, B. J.; Finke, R. G. Chem. Mater. 2004, 16, 139−150. (See also the addition/correction in Chem. Mater. 2004, 16, 3972. 1725
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