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Catalyst Sintering Kinetics Data: Is There a Minimal Chemical Mechanism Underlying Kinetics Previously Fit by Empirical PowerLaw Expressionsand if So, What Are Its Implications? Eric E. Finney*,† and Richard G. Finke*,‡ †

Department of Chemistry, Pacific Lutheran University, Tacoma, Washington 98447, United States Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, United States

‡

S Supporting Information *

ABSTRACT: Catalyst sintering is an undesired, but general, and hence practically important catalyst deactivation process. Understanding sintering kinetics and, then, the associated mechanism(s) is an important goal, one crucial to being better able to limit and otherwise control catalyst sintering rationally. However, and despite the availability of atomic-based sintering models, the kinetics of sintering of practical catalysts are to this day most often accounted for by curve-fitting with empirical power laws. Such empirical kinetics treatments are, unfortunately, devoid of rigorous mechanistic insight because they lack the balanced chemical equations that are required to define the rate constants and to also define the proper concepts and associated words that, in turn, are crucial for being able to describe correctly the sintering process physically. Hence, addressed herein is the key, previously unanswered question: is there a disproof-based, Ockham’s razor-obeying, hence mechanistically rigorous, minimal chemical mechanism that can be used to curve-fit sintering kinetics data previously accounted for by empirical power law expressions? If so, then what are its implications? The results provided demonstrate that literature catalyst sintering data, previously fit using empirical power laws, can instead be quantitatively accounted for by a simple, deliberately minimalistic, twostep kinetic model consisting of bimolecular nucleation of agglomeration, B + B → C (rate constant k3), followed by autocatalytic agglomeration, B + C → 1.5C (rate constant k4), in which B is the average starting nanoparticle, and C is the average larger, agglomerated nanoparticle. The results and findings compellingly demonstrate that the two-step mechanism can account for a variety of sintering kinetics data previously fit only by empirical power laws. Evidence is presented that the kinetic model appears to correspond to what has been called Particle Migration and Coalescence (PMC) in the prior literature. Ten conclusions and hypotheses, as well as four caveats, are listed in the Conclusion section, along with suggestions for further research.

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cost of the catalyst, and result in increased energy use.”35 Since sintering leads to a decrease in catalytic activity, many current studies are focused on preventing agglomerative sintering and on maintaining the particles in their active, metastable state. Understanding the kinetics and associated mechanisms of nanoparticle agglomerative sintering is crucial if one hopes to be able to rationally control and limit catalyst sintering. Such understanding begins with kinetics data that can be accounted

INTRODUCTION Metal nanoparticles are widely used as catalysts for a number of different chemical processes, including synthesis, energy conversion, catalytic chemical sensors, and applications in medicine and biology.1−7 The particles’ high surface energy due to their large surface area make them extremely active catalysts, but also leave them susceptible to catalyst sintering.8−33 Catalyst sintering can be defined as “that sum of processes where (i) catalyst agglomeration occurs with an accompanying loss of catalytic surface area and, hence, loss of activity; where (ii) collapse of the catalyst support occurs;34 or where (iii) both of these (and any other) processes occur that limit a catalyst’s lifetime, reduce a catalyst’s productivity, increase the net final © XXXX American Chemical Society

Received: June 29, 2017 Revised: August 11, 2017 Accepted: August 11, 2017

A

DOI: 10.1021/acs.iecr.7b02633 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

rate constants, and the allowed, proper concepts and words for describing those balanced reactions.49,50

for by well-defined, balanced chemical reactions that in turn define the rate constants and, importantly, also define rigorously the concepts and associated, proper words that one is then permitted to use to describe the sintering process physically. The Current State-of-the-Art in Fitting Sintering Kinetics Data. The presently most common treatment of sintering kinetics data of practical, nonmodel catalysts and systems is by curve-fitting to a “General Power-Law (GPL) Expression”.30 Indeed, a well-established expert in catalyst sintering atomistic models, Prof. C Campbell noted, in a 2002 Science paper, that “A ‘generalized power law’ is more commonly used now to model sintering kinetics, although it has no underlying atomistic mechanism”.16 In recent years, many kinetic models for catalyst sintering have been developed; some use the GPL expression or earlier work as a starting point,36 while others take an ab initio approach; still others have used computational simulation methods to describe particle migration and coalescence.37,38 Despite these advances in modeling catalyst sintering kinetics, many studies still rely on the GPL expression when it comes time to actually f it their kinetics data.39−42 For example, the main data examined herein are from recent, 2012 sintering kinetics literature, and were treated only by power-law empirical expressions, a fact which makes clear that such empirical power-law treatment of sintering kinetics data is still, unfortunately, the (crude) stateof-the-art. A common GPL expression is given in eq 1,26,30,43 although we note that small differences exist between different empirical GPL models in current use.40,42 In eq 1, D is the dispersion of the cluster (defined as the ratio of surface atoms to total atoms in the cluster), D0 is the initial dispersion, and Deq is the dispersion of the final cluster after sintering is complete. The empirical eq 1 also includes a poorly defined rate parameter kS, which is not a rate constant, because kS does not have a corresponding balanced chemical equation as is required to define individual rate constants. This alone means that kS contains no rigorous, unambiguous mechanistic meaninga huge, crucial point that by itself hints at the poor state of analysis of kinetics data in the catalyst sintering area. Without a reliable rate law, there is no reliable mechanism, and empirical power-law fits do not yield reliable, mechanistically useful, rate laws. Equation 1 also includes the exponent n, an integer whose value is proposed to be dependent on the mechanism of nanoparticle sintering: smaller values of n (≤4) are putatively associated with Ostwald ripening (OR), while larger values (≥5) are attributed to particle migration and coalescence (PMC)44−47 (although the exact distinction varies between reports). Again, this is ultimately mechanistic nonsense: the empirical parameter n cannot contain rigorous mechanistic insight except by accident or correlation with rate constants defined by balanced chemical equations. The significance of this point is hard to overstate. Worse, when a single value of n is unable to fit all of the data, those data are often separated and somewhat arbitrarily fit to two dif ferent values of n, the implication being that the mechanism has purportedly changed during sintering from OR to PMC or vice versa, for example.48 One need not look further to see the lack of mechanistic rigor and resulting confusion one has in trying to connect in any meaningful way such empirical parameters, kS and n, to the real, underlying, chemical, physical, and mechanistic processes of catalyst sintering. The missing tool here is the use of balanced chemical reactionsthat, in turn, define rigorously both the

d −

( ) = k ⎛⎜ D D D0

dt

s

⎝ D0

−

Deq ⎞ ⎟ D0 ⎠

n

(1)

In short, to avoid creating confusion about what is actually occurring physically during the sintering process and in order to deduce reliable mechanisms, one must have balanced chemical reactionsif not strictly elementary, then at least pseudoelementary steps. A pseudoelementary step51−54 is defined as a summation of elementary steps that can, however, still be used kinetically and mechanistically, a step that will still contain key kinetic features of the more detailed, true-elementary steps, atomistic mechanism (for an introduction to the concept of pseudoelementary reactions see the pioneering work of Professor Richard Noyes51−53 as well as our own studies54). Pseudoelementary steps and associated kinetic treatments are well-established as a key concept in deducing the mechanisms of more complex chemical systems,51−54 although this important point is underappreciated presently. The Present Disconnects in Attempts to Associate Empirically Treated Kinetics Data with Atomistic Sintering Studies and with the Macroscopic Sintering Concepts of Ostwald Ripening and Particle Migration and Coalescence. The use of empirical rate equations to account for practical catalyst sintering kinetics data is, unfortunately, common, despite the availability of detailed atomistic sintering models16,55−58 and despite two generally accepted sintering concepts. Those two concepts are Ostwald ripening (OR), in which larger clusters grow at the expense of smaller ones,59−63 and the generally accepted collection of phenomena referred to as particle migration and coalescence (PMC),9,22,43 in which nanoparticles diffuse, for example, across (or through) a (porous) support and then agglomerate to form larger particles. Throughout the literature, there is little agreement on which sintering process is operating for specific systems and specific conditions; it is often concluded that both processes operate at different times during the sintering process,44−47 although this conclusion is often reached solely based on the fact that neither model alone seems to be able to account for the observed data. Compelling kinetics data for this parallel pathway conclusion is typically missing, and confusion and intermixing of OR and PMC is common in the sintering literature. OR is the better studied of the two sintering concepts of PMC and OR, with the latter having atomic-level treatments and extensive experimental studies that appear to provide good support for the general OR picture.59−63 One curious, but telling, observation from a perusal of the catalyst sintering kinetics literature is that physical-chemical/ atomistic models that exist for catalyst sintering are rarely used to curve-f it and therefore account for practical catalyst sintering kinetics data (i.e., and other than by the builders of those specific models). This interesting observation is discussed more in the Supporting Information for the interested reader. The relevant point for the present study is that a simple, welldefined, mechanistically rigorously obtainedread, a disproofbased and Ockham’s razor obeyingkinetics model that others can, and will, routinely use to fit catalyst sintering kinetics data is badly needed to replace the widespread use of empirical power-law fits. B


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sintering kinetics data previously fit by empirical GPL kinetic models. But, first, a look at the four-step model and its key steps and associated features is necessary in order to understand why and how the last two steps of this model make sense to try as a minimalistic nanoparticle sintering kinetics model. The four-step kinetics model in Scheme 1 consists of the four pseudoelementary steps,64 beginning with a monomeric precatalyst denoted A in general kinetics terms, as kineticists prefer to write such kinetics schemes to keep them applicable to the general case. The four steps for nanoparticle formation and then the two types of agglomeration are, then: nucleation, A → B (rate constant k1), autocatalytic surface growth, A + B → 2B (rate constant k2), bimolecular agglomeration (alternatively describable herein as bimolecular nucleation of agglomeration (i.e., of agglomerative sintering), vide inf ra), B + B → C (rate constant k3), and lastly autocatalytic agglomeration (of the smaller B, and larger, C, nanoparticles), B + C → 1.5C (rate constant k4) (alternatively described in what follows as autocatalytic agglomerative sintering of smaller with larger nanoparticles). Note that there is stoichiometric counting in these balanced equations and even though B and C are, respectively, average smaller and larger nanoparticles: 1A nucleates to 1 larger cluster, B, in the first step, 1A and 1B grow to 2B in the second reaction (where B is a reactant and then also a product, in a 1:2 ratio, that reaction being the exact chemical definition of autocatalysis49,50), then 2 B agglomerate to 1 large particle, C, and then 1B and 1C agglomerate to what has to be (by the afore-defined stoichiometries) 1.5 C (since C contains 2B). Note this B + C → 1.5C step is also autocatalytic since C is both a reactant and a product formed in a greater than 1 amount. The sum of the B + B → C, then B + C → 1.5C pseudoelementary steps adds up to an overall stoichiometry of B + B + B → 1.5C (= B → 0.5C), so, again 2B equates to 1C. This four-step mechanism presently serves as the precisely balanced-equation-defined, and hence the mechanistically rigorously defined, presently most broadly applied, minimal chemical mechanism for nanoparticle formation, growth, and agglomeration.64−66 As noted above, this four-step mechanism is strongly supported by the disproof of over 21 alternative possible mechanisms en route to its formulation over nearly 2 decades.64−68 The four-step mechanism comprises our now extensively cited (>450 citations presently) 1997 “Finke− Watzky” (hereafter FW) two-step mechanism54 of A → B (rate constant k1), then A + B → 2B (rate constant k2), and then its 2014 bimolecular nucleation update68 (A + A → 2B and then A + B → 2B), as the solid foundation68 on which it was developed. These minimalistic mechanisms are currently widely usedto date the most widely used minimalistic kinetic scheme, for the analysis of sigmoidal kinetics data throughout nature by multiple researchers investigating a broad range of nucleating, growing, and/or aggregating systems. A partial list of specific cases and systems include, but are not limited to, Ir(0),64,69 Rh(0),70−73 Pt(0),64,74 Ru(0),75 Pd(0), Ni(0),76 Ag(0),77,78 and Au(0),79 nanoparticles, including direct XAFS support for the two-step mechanism in the cases of Rh(0)73 and Pt(0)74 nanoparticles, and direct SAXS support for the two-step mechanism in the case of Ni(0) nanoparticles.76 The two- or four-step models are also employed in homogeneous catalyst formation,80−82 heterogeneous catalyst formation,83−85 protein aggregation,86−89 solid-state kinetics,90,91 dye aggregation,92 and other areas of nature showing “cooperative”, autocatalytic phenomena such as, intriguingly, impurity uptake

Overall, then, the present state-of-the-art in describing the kinetics and associated mechanism(s) of catalyst sintering is, unfortunately, presently trifurcated into three areas that eventually need to be melded into one body of cohesive, coherent, knowledge about catalyst sintering: (i) valuable, detailed, physical-law containing, atomic-level studies of catalyst sintering from a variety of seminal works;16,44,55,57,58 but then (ii) the disconnected use, instead, of empirical power-laws to actually fit practical catalyst sintering kinetics data;39−42 and then, the even more insidious disconnect (iii) in which OR and PMC word-only descriptions of sintering9,59−63 are subsequently used to talk about the sintering process that was fit by the empirical power-law kinetics treatmenteven though those empirical models have no direct or rigorous connection to the words and concepts of OR and PMC, and even though the power-laws contain no rigorous mechanistic insight. This trifurcation of sintering knowledge is both little recognized yet a huge obstruction toward attaining the next level of understanding and reliable kinetic and mechanistic insight into practical catalyst sintering. It follows that melding this trifurcation into one cohesive body of knowledge represents another significant challenge, as well as an important research opportunity, in the area of catalyst sintering. The first step to bridging this trifurcation is to find a disproof-based, minimalistic, easily applicable kinetics model that can account for sintering kinetics data analyzed previously by empirical power-law treatments. Just such a kinetics model for catalyst sintering is the main focus of the present contribution. The Four-Step Mechanism for Nanoparticle Formation and Agglomeration that Includes Two Agglomeration Steps that Will Prove Useful for Quantitatively Fitting Sintering Kinetics Data Previously Described by GPL Empirical Treatments. The best studied and presently most widely applied (four-step), deliberately minimalistic kinetic model for nanoparticle formation and agglomeration in Scheme 1 is composed of nanoparticle nucleation, autocatalytic growth, and importantly for this work, two types of agglomeration.64−67 The last two steps of this highly disproofbased (>21 alternative kinetics schemes disproved to date64−66), Ockham’s-razor-obeying, again deliberatively minimalistic kinetics model are the two pseudoelementary steps that we will demonstrate herein are able to account for catalyst Scheme 1. Four-Step, Double Autocatalytic Mechanism for Nanocluster Nucleation, Autocatalytic Growth, And Then Two Types of Agglomerationa

a

One pictorial representation is given to the right. Note that a larger Kinetically Effective Nucleus68 (to be distinguished from a so-called “critical” nucleus68) is shown to the right of the first step, as a result of the implied, composite steps of nA → nB = Bn. However, recent evidence68 reveals that n can be smaller than depicted, perhaps often just n = 2,3.68 C


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Industrial & Engineering Chemistry Research by resins used for water purification.93 There is also a three-step model that includes just the first three steps of Scheme 1.94 Together the two-, three-, and four-step kinetic models have over 800 citations at present. An obvious starting hypothesis based on considerable precedent (vide supra), then, is that the two agglomeration steps that are the third and fourth steps of the four-step mechanism hold promiseperhaps great promise based on the findings hereinfor initial curve-fitting analyses of catalyst sintering kinetics data. An alternative hypothesis to be disproved, however, is that despite the good fits to catalyst sintering kinetics data, the underlying physical process should not behave intrinsically as a nucleated process, or that autocatalytic agglomeration is not a physically reasonable process for catalyst sintering. The results obtained and literature precedent will be used to test this alternative hypothesis. A second alternative, and hence overall third, hypothesis is that an alternative minimalistic two-step kinetic model (that looks more like OR, vide inf ra) might also be able to fit the kinetic sintering data examined herein that has been traditionally fit by power-law expressions. This alternative hypothesis is also addressed in what follows. An Initial Bridge for at Least Two of the Components of the Trifurcation: A Disproof-Based, Ockham’s-RazorDerived and Hence Obeying, Minimal Chemical Mechanism for Fitting Practical Catalyst Sintering Kinetics Data. Provided below in Scheme 2 is a deliberately

constant nomenclature. The rate constant nomenclature of k3 and k4 (and not k1 and k2) are analogously used in Scheme 2, again to be consistent with the prior literature.64,90 A Recent, First Application of the Two-Step Sintering Kinetic Model in Scheme 2 to One Initial Set of Catalyst Sintering Kinetics Data. Recently we reported the first evidence that the two-step mechanism in Scheme 2 is able to serve as a quantitative description of sintering kinetics data, in that case sintering kinetics observed for a very well-defined, initially atomically dispersed, Ir0/zeolite catalyst,35 Scheme 3. In Scheme 3. Growth and Sintering of Ir Clusters Dispersed in Zeolite Y35

that system from Prof. B Gates’ laboratory and in a collaborative investigation, the atomically dispersed Ir01 catalyst (correctly labeled35 as B, since it is preformed by H2 reduction of a monomeric, IrI1 precursor, A)35 underwent three successive cycles of 3800 turnovers of cyclohexene hydrogenation at the relatively mild temperature of 72 °C. Over the course of these catalytic cycles, the Ir1 catalyst sintered to Ir4−6, Ir40−50, and then Ir∼1600 (the latter observed outside the zeolite pores), as determined by aberration-corrected HAADF-STEM and EXAFS.35 The diameter vs time kinetics data for the growth of the Irn catalyst during the catalytic cycles was quantitatively fit by the two pseudoelementary steps defined in Scheme 2 of bimolecular nucleation of agglomeration, then autocatalytic agglomeration between the smaller (B) and larger (C) nanoparticles. The autocatalytic agglomeration, k4, process proved to be faster than the k3 process, as we have seen more generally before for solution-based nanoparticle agglomeration (“sintering”).35,64−66 However, the preliminary studies applying Scheme 2 and eq 2 to just this one, initially atomically dispersed, sintering system left many important questions unanswered. How general is the ability of eq 2 to fit sintering kinetics data? Can eq 2 be used to fit a broader range of sintering data commonly fit by GPL empirical treatments that presently dominate the treatment and curve-fitting of literature sintering kinetics data? The twopseudoelementary-step kinetics model in Scheme 2 looks on the surface like it could be the first description of the minimal chemical mechanism underlying so-called Particle Migration and Coalescence, since both steps physically involve “particle migration, then coalescence”. However, the question arises of if there is also a minimalistic two- or three-step model for Ostwald Ripening that could be tried as an alternative, minimalistic kinetics model that might also be able to account for empirical power-law kinetic fits? It is precisely these and other key questions that are addressed in what follows. Finally, a couple of important notes merit mentioning before concluding this deliberately complete Introduction and Background section. One is that the pseudoelementary and other approach herein, as well as the results obtained, parallel closely conceptually our 2009 report95 in which we applied our twostep, A → B, A + B → 2B nucleation and growth mechanism to,

Scheme 2. Deliberately Minimal, Two-Step, Mechanism for Transition-Metal Nanoparticle Bimolecular Nucleation (of Agglomeration), Followed by Autocatalytic Agglomeration and Associated eq 264,90a

a

The associated diameter vs time equation35 is also provided as eq 2; note the difference between the nanoparticle diameter, D, eq 2, vs the nanoparticle dispersion, D (i.e., in italics), as defined back in eq 1.

⎛ ⎞1/3 4k 3(1 − ek4[B]0 t /2) 3 3 3 Dt = ⎜⎜D0 + (Df − D0 )⎟⎟ 4k 3(1 − ek4[B]0 t /2) − k 4 ⎝ ⎠ (2)

minimalistic kinetic model that we demonstrate herein can quantitatively account for literature sintering kinetics data previously accounted for by an empirical GPL treatment. The proposed two-step kinetic model in Scheme 2 consists of the two steps, B + B → C and then B + C → 1.5C. Note that it is important to distinguish conceptually the two-step sintering mechanism proposed in Scheme 2 of B + B → C (rate constant k3) and then B + C → 1.5 C (rate constant k4) from the twostep mechanism with bimolecular nucleation68 of A + A → 2B (rate constant k1) and A + B → 2B (rate constant k2). Note that the starting material in Scheme 2 below is conceptually correctly labeled as B (and not “A”), since catalyst sintering starts with (in that case) preformed particles, B, not the nanoparticle precursor, A, which is often a monomeric species. The agglomerated, larger average nanoparticle is defined as C, again to be consistent with the prior literature64,90 and its rate D


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term [B]0 (that, for supported catalysts might better be denoted a standard state, defined for example as mol B/surface area, as discussed more in the Supporting Information), and because no “concentration” or other such data are given along with the literature data employed, for the purposes of fitting the data we have employed a standard state definition of [B]0 = 0.10 M. Importantly, fitting the data to eq 2 using different values of [B]0 as controls does not affect the quality of the fit, nor the statistics from the fits, nor does it change the relative values of the rate constants k3 and k4. However, the absolute values of k3 and k4 depend, as expected, on the standard state value chosen for [B]0 (see the Supporting Information for supporting data and additional discussion of this point). Hence, in what follows conclusions are based only on the relative statistics and quality of the fits and the relative k4/k3 values. The absolute k3 and k4 values cited refer to the standard state chosen of [B]0 = 0.10 M. As discussed more in the Supporting Information for the interested reader, the experimentally important corollary here is that authors wishing to use eq 2 to fit their own sintering kinetics data need to report [B]0 values in terms of their chosen standard state. Statistical judgment of the quality of the fits was performed in two ways. The coefficient of determination, R2, was calculated using eq 3.

in that case, solid-state phase transformation kinetics that historically had been described using semiempirical “dispersive kinetics” such as the Avrami−Erofe’ev equation or one of its greater than 15 variants.95 That work has a number of parallels to the present work that also strives to undo the confusion produced by using empirical or semiempirical equations that lack mechanistic meaning to try to account for (in that case) solid-state kinetics data (see the indicated reference95 as well as the Supporting Information of the present contribution for additional details and discussion). Second, highly relevant to the present study and hence recommended reading is the discussion in another paper90 containing the strengths and 10 previously hidden physical insights which result from the two- and four-step mechanisms back in Scheme 2 and the weaknesses of the two- and four-step mechanisms. The weaknesses ultimately derive from the oversimplified nature of these deliberately, initially minimalistic models, and the resultant fact that they yield only average insights about average particles B, C, and average rate constants, k3 and k4. The Supporting Information elsewhere90 also contains recommended reading covering references, discussion, and epistemology regarding the construction of different types of models in science, notably disproof-based, minimal mechanistic models, physical-chemical based models, and engineering models, three types of dynamic models that differ in their adherence (or not) to Ockham’s razor and, concomitantly, their “allowed” number of parameters (that, in general, increase, from mechanistic models to physical-chemical to engineering models) as well as if each model includes disproof of other models, or not (levels of disproof which, generally, decrease from mechanistic to physical-chemical to engineering models). Third and lastly, the close parallel of the present studies with parts of our prior work90,95 are worth noting and will prove of interest in understanding the value and broader generality of the effort and findings which follow. Those90,95 as well as the present studies have in common the goal of replacing empirical expressions by balance-equation-based, rigorous kinetic and mechanistic schemes, in turn providing well-defined concepts and words able to provide meaningful physical insights into the chemical, kinetic, and mechanistic processes at hand.

R2 = 1 −

SSres SStot

(3)

∑iN= 1(yi

The residual sum of squares, SSres, is given by SSres = − f i)2, where y is the experimental value (of the diameter herein) and f is the value predicted by the model. The total sum of squares, SStot, is given by SStot = ∑i N= 1(yi − y)̅ 2, where y ̅ is the average of all of the data values. Values of R2 closer to 1 indicate a closer correlation and therefore a better fit for the model. The other statistical method used to judge the fits was Akaike’s Information Criterion (AIC),90 which determines the probability that one model is better than another at fitting the data. Specifically, we used Akaike’s second-order method90 to calculate values of AICc for the two equations: ⎛ SS ⎞ 2K (K + 1) AICc = N ln⎜ res ⎟ + 2K + ⎝ N ⎠ N−K−1

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(4)

In eq 4, N is the number of data points, SSres is the residual sum of squares as defined above, and K is the number of parameters in the model. Using this equation, lower values of AICc correspond to better fits, and the difference between AICc values can be used to determine the probability that one model should be used rather than another.

EXPERIMENTAL SECTION Selection of Literature Sintering Kinetics Data. We began with literature data for the sintering of a model Pt/SiO2 nanocluster catalyst reported in 2012 by Larsson and coworkers,12 the kinetics of which were followed by transmission electron microscopy (TEM) and by indirect nanoplasmonic sensing. These kinetics data were selected for fitting for a number of reasons. First, their report is a prototype example of one that explicitly uses the phenomenological eq 6 (vide inf ra) to fit the data. Second and more importantly, this study provides many sets of high-quality kinetics data, taken at a number of different temperatures. The fits to sintering at different temperatures allow us to examine the effect of temperature on both the empirical equation parameters and on the autocatalytic agglomeration rate constants. Data Digitization, Curve-Fitting, Choice of Standard State, and Statistical Analysis. The data from the above report were digitized using the free, online tool Web Plot Digitizer (http://arohatgi.info/WebPlotDigitizer/). Fits to the data to eqs 2 and 6 (see below) were performed using the freeware CurveExpert. Because eq 2 includes a “concentration”

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RESULTS AND DISCUSSION Pt/SiO2 Sintering Kinetics Data. Larsson and co-workers fit their Pt nanocluster size vs time kinetics data to the empirical equation for average particle radius, where A and n are “adjustable parameters”.12 dR A = n−1 dt R

(5)

The integrated rate law is then R t = (R 0n + Ant )1/ n

(6)

For the Pt/SiO2 system, size vs time data obtained from TEM measurements are given for sintering at 550 °C (Figure 3 therein12). These six data points give a curve that can be fit E


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Industrial & Engineering Chemistry Research using both equations so that the results of the two fits can be directly compared (Figure 1).

Figure 3. Sintering kinetics data (diameter vs time) at 500 °C taken from ref 12, Figure S2 of the SI therein. The fits are to eq 2 (two-step mechanism, blue curve) and eq 6 (power law, red curve). For the fit to eq 2, k3 = 5.39 × 10−4 M−1 min−1 and k4 = 9.37 × 10−3 M−1 min−1 (for [B]0 = 0.10 M); R2 = 0.9610, AIC = −732. For the fit to eq 6, A = 1.65 × 10−4 and n = 4.19, R2 = 0.9575, AIC = −724.

Figure 1. Nanocluster size vs time data for the growth of Pt/SiO2 as measured by TEM (and taken from Figure 3 elsewhere12). The fits are to eq 2 (two-step mechanism, blue curve) and eq 6 (power law, red curve). For the fit to eq 2, k3 = 0.0138 M−1 min−1 and k4 = 0.0532 M−1 min−1 (for [B]0 = 0.10 M); R2 = 0.9369; AIC = −21.7. For the fit to eq 6, A = 0.0466 and n = 6.74; R2 = 0.9431; AIC = −23.7.

Sintering data at 550 °C, and at three additional temperatures, were obtained by Larsson and co-workers using realtime indirect nanoplasmonic sensing (INPS) spectroscopy.12 Those data, found in the Supporting Information of the original report,12 were digitized and fit to eq 2 and eq 6. Those fits are given in Figures 2−5.


Figure 2. Sintering kinetics data (diameter vs time) at 450 °C taken from ref 12, Figure S2, of the SI therein. The fits are to eq 2 (two-step mechanism, blue curve) and eq 6 (power law, red curve). For the fit to eq 2, k3 = 1.13 × 10−3 M−1 min−1 and k4 = 9.46 × 10−3 M−1 min−1 (for [B]0 = 0.10 M); R2 = 0.9493, AIC = −663. For the fit to eq 6, A = 0.882 and n = 22.7, R2 = 0.9621, AIC = −686.

Statistical Analysis of the Data. The statistics of each fit are summarized in Table 1. Given in this table are the coefficients of determination (R2) and the Akaike’s criterion values (AICc) for each fit. From the AICc data, we can obtain the Akaike weight, w, which is defined as w=

e−ΔAICc/2 1 − e−ΔAICc/2


(7)

The value of w is the relative probability that the given model is favored. To further quantitate this probability, we can F


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Table 1. Statistical Data for the Fits in Figures 2−5 to eq 2 and eq 6. R2 Closest to 1 and Most Negative AICc Represent the Best Fit temp 450 500 550 610

eq 2 R2 0.9493 0.9610 0.9796 0.8801

eq 2 AICc −663 −732 -647 −955

eq 6 R2

eq 6 AICc −686 −724 −670 −988

0.9621 0.9575 0.9843 0.8913

determine the so-called “evidence ratio” (ER), the ratio of w values. All of these statistical data are presented in Table 1. As can be seen both visually and from the statistics, the autocatalytic agglomeration and the empirical power-law equation both fit the data very well. The differences in R2 values are very small in each case, indicating that the quality of the fits for each model is not significantly different. Of note is the much larger apparent difference in the quality of the fits when looking at the Akaike weights, w. Previously, we observed that evidence ratios of 10±4 were not significant with regard to choosing a better model;95 in other words, when the evidence ratio is within 10±4, both models can be considered to fit the data equally well.90 Given that each ER value in Table 1 falls either within or just outside this range (which itself is somewhat arbitrary; ER values for similar systems have reached90 10−184), statistically there appears to be little to no difference in the quality of the fits between the autocatalytic agglomeration mechanism and the empirical power-law rate equation used by Larsson and many others. In light of the similarity of the fits statistically, we wanted to see if eqs 2 and 6 could be considered somehow equivalent.90 To do this, we expressed both equations in similar forms and compared them. The derivation and insights that can be gleaned from this analysis are provided in the Supporting Information. Analysis of Other Literature Systems. Examination of other literature systems that study nanocluster sintering reveals further evidence in support of the two-step sintering mechanism as a minimal chemical mechanism for kinetics previously fit by empirical power-law expressions. For example, Simonsen et al. studied the oxygen-induced sintering of Al2O3supported Pt clusters by TEM measurements.8 These authors attribute the sintering of clusters to an Ostwald ripening mechanism, in which eq 8 is followed: ⎞ dR α⎛ R = 2⎜ − 1⎟ ⎝ ⎠ dt R R*

w (eq 2) −6

9.65 × 10 0.9842 8.62 × 10−6 6.21 × 10−8

w (eq 6) 1 1.58 × 10−2 1 1

ER (eq 6/eq 2) 1.04 1.61 1.16 1.61

× × × ×

105 10−2 105 107

Figure 6. Kinetics data (diameter vs time) for the sintering of Al2O3supported Pt clusters as measured by TEM, taken from Figure 6a of ref 8. The fits are to eq 2 (two-step mechanism, blue curve) and eq 6 (power law, red curve). For the fit to eq 2, k3 = 7.13 × 10−5 M−1 min−1 and k4 = 7.20 × 10−1 M−1 min−1 (for [B]0 = 0.10 M); R2 = 0.9926, AIC = −30. For the fit to eq 6, A = 2.97 × 10−3 and n = 4.26 × 10−3, R2 = 0.8874, AIC = −6.

This qualitative description sounds like what one expects for the bimolecular nucleated and then autocatalytic agglomeration sintering mechanism, so we tried to fit the data using eq 2 as well as the power-law in eq 6. Equation 2 corresponding to the two-step mechanism clearly gives a superior fit; the empirical power-law equation does not appear to be able to fit the sigmoidal nature of the diameter vs time data. The inability of eq 6 to adequately describe the kinetics of this system is also evident from the small, noninteger, nonphysical value of n (a (“dimensionality” of n = 4.26 × 10−3), which is far from the typical values of 3−6 more typically observed when the powerlaw equation is used. More recently, a TEM study of the growth of Pt nanoparticles supported on Al2O3 at temperatures of up to 800 °C was carried out by Zhang et al. with the purpose of distinguishing a PMC growth mechanism from Ostwald ripening.44 They interpreted the sigmoidal growth kinetics (Figure 7) as evidence of the two mechanisms occurring sequentially. However, the superior fit of the data to the single mechanism represented by eq 2 strongly supports an alternative hypothesis: that the sigmoidal growth rate is due to a bimolecular nucleation, followed by autocatalytic agglomeration, a single two-step mechanism and not some convolution of two (unknown) mechanisms. Moreover, that two-step mechanism at least looks more like PMC rather than Ostwald ripening, given that each of the B + B → C and then B + C → 1.5C steps employed involves “particle migration and coalescence”, although we will return later to the question of whether or not the two-step mechanism should be associated with the term “PMC”. In short, the data and fits in Figure 6 further support the hypothesis that what is currently attributed44 to two different sintering mechanisms is really just two distinct steps of the same mechanism, each with their own physically meaningful rate constants, specifically the steps of

(8)

Here, α is “a system-dependent parameter” and R* is the “critical radius”, defined as the radius at which a particle becomes stable (i.e., where growth or shrinking are equally likely) and following the use of the “critical radius” in classical nucleation theory.96−98 (This is similar to the reworked definition Datye and co-workers used in their definition of the “critical radius” in nanoparticles undergoing OR, namely “the critical radius is the size of the particle that is in equilibrium with the surrounding adatom concentration. Such a particle neither shrinks nor grows due to OR...”.25) The cluster diameter vs time data, reproduced in Figure 6, clearly show an initial period of slow growth followed by rapid growth; the authors themselves describe the process as “the mean diameter remains almost stable during the initial period, but suddenly it increases rapidly, and subsequently it becomes roughly constant with time”.8 G


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“turn on” the often faster, autocatalytic agglomeration step, B + C → 1.5C. The difference between the nucleation and autocatalytic growth steps can be seen by comparing their associated rate constants, k3 and k4. Just such a comparison is done for all of the data analyzed herein in Table 2. In each case the autocatalytic agglomerative sintering rate constant, k4, is greater than the bimolecular nucleation of sintering rate constant, k3. This can be seen qualitatively in each of the fits above and the resultant k3 vs k4 rate constants. The difference in k3 vs k4 is more pronounced in the data for sintering of the atomically dispersed Ir1/zeolite catalyst that we reported previously, the autocatalytic rate constant being ∼160 times greater than the nucleation rate constant for that system.35 An even more extreme example comes from the size vs time data from Simonsen et al.8 (Figure 6), where the k4/k3 ratio is 1.0 × 104; that is, once the autocatalytic sintering step has been “turned on”, the effective sintering rate constant increases by a factor of ten thousand in the k4, autocatalytic agglomeration step. Possible reasons for the relatively high values of k4 have appeared as part of our four-step mechanistic studies, notably the finding that surface-ligand to larger particle, C, bond dissociation energies appear to be lower than those for the smaller B particles, making C more coordinatively unsaturated and, overall, more reactive in at least many cases examined to date.64 This finding from the two-step mechanism fits, that sintering is a nucleated, autocatalytic process with k4 > k3 raises the question of whether or not this makes physical sense, the question and associated primary alternative hypothesis that it does not being examined next. A Return to the Alternative Hypothesis That, Despite the Good Fits, the Underlying Physical Process Should Not Intrinsically Behave as a Nucleated Process, or That Autocatalytic Agglomeration Is Not a Physically Reasonable Process for Catalyst Sintering. Does the finding that nanoparticle sintering is a bimolecularly nucleated, autocatalytic process, make physical sense? Does it make physical sense that the k4 step is faster than the k3 step? What is, then, the bulk of the evidence for or against catalyst sintering being a bimolecularly nucleated, autocatalytic process? The evidence so far is (i) that a range of kinetics herein (and the atomically dispersed, initially Ir(0)1 system studied before35) are well-fit by the two-step mechanism; along with the fact (ii) that no prior minimal chemical mechanism is presently known, certainly none with just two rate constant parameters, that can account for the range of sintering kinetics data with statistically equally or better fits. There is also good evidence already from the four-

Figure 7. Kinetics data (diameter vs time) for the sintering of Al2O3supported Pt clusters as measured by TEM, taken from Figure 4 of ref 44. The fits are to eq 2 (two-step mechanism, blue curve) and eq 6 (power law, red curve). For the fit to eq 2, k3 = 2.39 × 10−2 M−1 min−1 and k4 = 3.55 × 10−1 M−1 min−1 (for [B]0 = 0.10 M); R2 = 0.9917, AIC = −28. For the fit to eq 6, A = 9.91 × 10−2 and n = 7.37, R2 = 0.9751, AIC = −22.

bimolecular nucleation of sintering followed by autocatalytic agglomerative sintering. This example teaches the mechanistic insight that can occur even from the application of the present, deliberately minimalistic two-pseudoelementary-step mechanism for catalyst sintering, Scheme 2, vide supra. This example also teaches how the use of empirical fitting equations can quickly lead to questionable, if not incorrect, mechanistic conclusions such as the presence of two difference sintering mechanisms as in the case of Ptn nanoparticles supported on Al2O3.44 Two additional literature systems have been fit using eq 2 and eq 6, specifically the sintering of alumina-supported Pd clusters used in automotive catalysts99 and the sintering of alumina-supported Ni nanocluster reforming catalysts.100,101 The good fits to eq 2 given in the Supporting Information in Figures S3 and S4 demonstrate the broader generality of the two-step mechanism for fitting nanocluster sintering kinetics data. A Tabulated Summary of the Kinetics data and Fits Using the New, Two-Step Sintering Mechanism and Associated eq 2. A significant insight from the B + B → C, B + C → 1.5C mechanism is the hypothesis that nanoparticle sintering in the literature systems re-examined herein can be described as a nucleated, autocatalytic process. That is, just a little of the smaller B particles agglomerating to C, B + B → C, looks to be able to

Table 2. Summary of the Rate Constants k3 and k4, as Well as Their Ratios k4/k3, and Empirical Rate Parameters A and n for Each Set of Data Analyzed Herein system Pt/SiO2 TEM data at 550 °C (Figure 1) Pt/SiO2 at 450 °C (Figure 2) Pt/SiO2 at 500 °C (Figure 3) Pt/SiO2 at 550 °C (Figure 4) Pt/SiO2 at 610 °C (Figure 5) Pt/Al2O3 (Figure 6) Pt/Al2O3 at 800 °C (Figure 7)

k3 (M−1 min−1)a 1.38 1.13 5.39 1.91 7.53 7.13 2.39

× × × × × × ×

−2

10 10−3 10−4 10−3 10−4 10−5 10−2

k4 (M−1 min−1)a 5.32 9.46 9.37 5.28 1.84 7.20 3.55

× × × × × × ×

−2

10 10−3 10−3 10−3 10−2 10−1 10−1

k4/k3b 3.85 8.37 17.4 2.76 24.4 1.01 × 104 14.85

A 0.0466 0.882 1.65 × 7.00 × 3.25 × 2.97 × 9.91 ×

n

10−4 10−2 10−3 10−3 10−2

6.74 22.7 4.19 12.8 4.64 4.26 × 10−3 7.37

These fits refer to a standard state of [B]0 = 0.10 M as detailed in the Experimental section. See the SI for discussion of the dependence of the absolute value of k3 and k4 on the initial [B]0. bThe ratio k4/k3 is, however, independent of the initial [B]0 as the data and discussion in the SI demonstrate. a

H


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Industrial & Engineering Chemistry Research step mechanism,64 discussed back in the Introduction, as to why it is physically reasonable to describe sintering as a nucleated, autocatalytic process, namely, (iii) the 21 alternative mechanisms disproved en route to the four-step mechanism, from which the two-step sintering mechanism is derived, mean that the two-step sintering mechanism employed herein comes f rom a highly disproof-based, rigorous mechanistic foundation. Additionally, (iv) the greater reactivity of C over B for agglomeration implied by the finding of k4 > k3 is consistent with and precedented by our earlier work on the four-step mechanism where the on-average larger, C, nanoparticles were, surprisingly at the time, discovered to be the faster hydrogenation catalysts than the smaller, B, nanoparticles.64 This observation alone implies logically that the k4, B + C → 1.5C step should be faster than the k3, B + B → C step, as is observed. Moreover and as first presented elsewhere,64 (v) the only reasonable hypothesis we have been able to come up with for why on-average larger nanoparticles, C, are more reactive catalysts than the smaller, B nanoparticlesand now why the same pattern is seen with k4 > k3is that C must be more coordinatively unsaturated, a hypothesis supported by a seminal paper in the nanoparticle literature revealing 2-times lower M(0)n-ligand binding enthalpies in bulk M(0)m (in that case Ni(0)n with N2 as the ligand) as compared to Ni(0)n nanoparticles (although there is the expected enthalpy/entropy compensation effect when one also looks at the entropy of N2 binding as a function of particle size).102 Furthermore, (vi) the finding that ΔH⧧4 > ΔH⧧3 for at least solution aggregating (“sintering”) nanoparticle systems examined so far103 is fully consistent with the finding of k4 > k3 at the higher operating temperatures of sintering systems examined herein, with (vii) the B + C → 1.5C step being dissociatively activated.103 (The relevant data here are the activation parameters for two-step fits of the agglomeration of Ir(0)n nanoparticles stabilized by a P2W15Nb3O689− polyoxometalate and n-Bu4N+ counter-cations in acetone: ΔH3⧧ = 6.2(3) kcal/mol, ΔS⧧3 = −46(2) e.u., and ΔH⧧4 = 18(1) kcal/ mol, ΔS⧧4 = −2.5 (2) e.u., in turn suggesting that the k3 step appears to be associatively activated, while the k4 step looks to be dissociatively activated, as if the k4 step involves a, say, speculative C-(ligand)a ⇌ C-(ligand)a-b + b ligand, prior equilibrium for the larger C particles.103) Of additional interest here is (viii) that faster agglomeration of larger and smaller particles has been predicted theoretically (due to their larger effective cross section) and claimed experimentally (albeit without quantitation) in polymer and sol−gel systems,104,105 but remained unverified until our discovery and quantification of the k3 and k4 rate constants, including their relative magnitude.35,64,103 Next, (ix) evidence for bimolecular (or other low-molecularity) nucleation is rapidly increasing for selfassembly growth or aggregating systems across nature,68 results that support the finding herein of good fits to the two-step mechanism involving bimolecular nucleation of autocatalytic agglomerative sintering. Finally, (x) there is the fact and associated broader precedent that, for nucleating, growing, and aggregating systems throughout nature, the two- and four-step mechanisms enjoy over 800 citations at present. In that sense, it is not unexpectedbut, instead, is well-precedented in a broad and general waythat the two-step sintering mechanism employed herein should do, as it does, an excellent job of accounting for sintering kinetics data previously fit only by empirical power laws. Analysis of Additional Sintering Literature of Special Interest in Light of the Two-Step Mechanism: Testing

Predictions of the Bimolecularly Nucleated, Autocatalytic Agglomeration, Minimal Sintering Mechanism. A look back at the sintering literature, in light of the results presented so far, provides additional, independent evidence in support of the two-step sintering, minimal mechanism. First, a prediction of the two-step mechanism is that unimodal size particles should sinter significantly more slowly due to the lack of the different particle sizes required for the faster, k4, B + C → 1.5C step of the two-step mechanism. Important literature evidence supporting just this point is available in a 2014 literature study showing that, indeed, monomodal Pt 68 nanoparticles undergo unusually slowly agglomerative sintering under argon compared to a mixture of Pt68 and Pt22 particles or to a broader distribution of Ptn≥53 particles.106 Although the data in that valuable study were interpreted in terms of an OR (Ostwald ripening) process,106 the two-step mechanism provided herein offers a ready explanation for the observed data, and in turn raises the question of if the prior assignment of OR is actually correct. On reflection, one realizes that the high sintering resistance seen for the smaller Pt22 particles106 is actually evidence against OR, which postulates that smaller, more energetic particles should sinter faster by providing, for example, a facile Pt22 to Pt21 + Pt1 equilibrium, the Pt1 + Pt22 then giving Pt23, and so on as the basic OR mechanism postulates.59−63 The fact that sintering of Pt68 is accelerated by the presence of small amounts of larger Pt136 (formed by dimerization of Pt68) is once again opposite of what is expected for OR in which smaller particles are expected to enhance sintering.55,107 Hence, if the two-step mechanism actually is a balanced chemical equation version of “PMC”, then this is an important example where the mechanistic conclusion in the literature that OR is occurring appears to be wrong and, hence, merits re-examination. Additionally, the transient bi- to trimodal particle size distributions (PSDs) seen when sintering Ptn on Al2O3 or, separately, on two types of SiO2 under 4% O2 at 500, 550, and 600 °C108PSDs that did not match the PSDs predicted by OR or classic PMC108can also be explained by a nucleated process that is part of the two-step sintering mechanism. More specifically, the effect of surface heterogeneity of the oxidesupport on the observed PSDs108 is readily explained by the hypothesis of those support effects operating primarily on the B + B → C nucleation step, effects that are then amplified by the autocatalysis of the B + C → 1.5C step and the fact that k4 is often > k3. Hence, the weight of the evidence is strong that bimolecular nucleation of subsequent autocatalytic agglomeration is a physically reasonable way to describe catalyst sintering. The observant reader will note that none of the above evidence nor insights have resulted previously from the application of powerlaw or other prior kinetic treatments or models of catalyst sintering despite the at least ≥ 40 year history of studies of catalyst sintering.26,57,58 Is There a Different, Minimal OR-Based Mechanism That Can Also Fit Power-Law Sintering Kinetics Data? Hence, Does the Two-Step Mechanism Also Include OR, or if Not, Then Is the Two-Step Mechanism the Minimal Chemical Mechanism Underlying PMC? These questions are raised not only by the section above, but also and first by the alternative hypothesis in the Introduction, namely that formulation of a minimalistic two-step mechanism for OR might be possible and, if so, then conceivably might also fit sintering kinetics data presently accounted for historically by I


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Industrial & Engineering Chemistry Research empirical power-law expressions, perhaps even better than the two-step (ostensibly PMC, vide inf ra) mechanism presented so far. Hence, given in Scheme 4 is the best of three (vide inf ra; see also the Supporting Information) plausible, largely minimalistic Scheme 4. One Plausible, Largely Minimalistic Ostwald Ripening (OR) Mechanism Consisting of Dissolution of Smaller Clusters B into Monomers A and then Transport of Atoms to, and Addition to, Larger Clusters C Followed by Autocatalytic Addition of Monomers A to C

OR mechanisms we could come up with for attempted fitting of the sintering kinetics data. That three-step OR mechanism postulates an initial nanocluster B dissolving/shedding atoms, A, which then combine with both B, A + B → C, and in a separate step with the larger cluster product C, A + C → 4/3C, a formally autocatalytic step (since C is both a reactant and product formed in >1 equiv). The autocatalytic step was deliberately added so that this particular, best-of-three OR mechanism might have the best chance to fit sigmoidal data such as that shown first back in Figure 6. (The factor of 4/3 in the three-step OR mechanism results since the prior steps of B giving 2A, and then A + B giving C, in turn defines C stoichiometrically as containing 3A.) Note that as formulated, this three-step OR mechanism has the advantage of four parameters with which to fit the data, twice as many rate constants as the two-step mechanism back in Scheme 2 with its two rate constants. The Supporting Information presents the required conversion of D (diameter) to cluster concentration [C] for the three-step OR mechanism that is needed to try to fit the D vs time kinetics data with the postulated OR mechanism. The resulting equation was used, using numerical integration via the freeware Copasi, to fit two sets of kinetics data already shown in Figures 2 and 6, but now using the three-step OR mechanism in Scheme 4 and as shown in Figure 8. The results of the attempted fits to the two data sets are interesting. They reveal that, while this three-step, 4 variables OR mechanism can fit the data to some extent, it does not fit the data as well as the two-step sintering (ostensibly the “twostep PMC”, vide inf ra) model already described and previously used to provide good fits to this same data, Figures 2 and 6, vide supra, despite the two-step model having half as many rate constant variables. Attempted fits to other alternative OR mechanisms are given in the Supporting Information for the interested reader (Figures S5, S6, S7, and S8). These models provided even less satisfactory fits than did the three-step OR model in Scheme 4, particularly to the sigmoidal kinetics data shown in the bottom of Figure 8. Hence, the two-step mechanism back in Scheme 2 provides a superior accounting of the sintering kinetics data than does at least the three-step OR and the two other minimalistic OR mechanisms considered as part of the Supporting Informationeven though those OR mechanisms have more variables (a point we rechecked by doing the fit shown in Figure 8b

Figure 8. Attempted fits of the three-step OR model in Scheme 4 to (top) the data from Larsson12 for Pt/SiO2 at 450 °C, and (bottom) to the sintering of Al2O3-supported Pt clusters. For the top, Pt/SiO2 data, R2 = 0.9553; for the Pt/Al2O3 data, R2 = 0.8262.

multiple times, which returned the same fit shown each time). It in turn appears that the two-step mechanism provided herein is both (i) a minimal chemical mechanism underlying at least the catalyst sintering data considered herein (and perhaps even the minimal mechanism, since we at least have not been able to conceive of a simpler mechanism that can fit the data); and maybe also (ii) the minimum chemical mechanism for what has been previously expressed primarily in words as Particle Migration and Coalescence (PMC), what we hereby term as the “two-step PMC” mechanism. Arguing in favor of the “twostep PMC” mechanism is that it is just the last two steps of the four-step mechanism presented back in Scheme 1 for nanoparticle nucleation, growth, and two types of agglomeration, a highly disproof-based mechanism.64−68 Nevertheless, the “two-step PMC” mechanism should be considered primarily as a hypothesis going forward, one requiring further testing and attempts to disprove it, or if that disproof proves impossible, to thereby further support it. Resultant Mechanism-Based Insights into How To Control Catalyst Sintering. With the “two-step PMC” mechanistic hypothesis in hand, we can use its associated insights and predictions to propose how one can minimize undesirable sintering. The implications from the two-step J


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reasonable alternative hypotheses in Chamberlin’s classic words109have been considered and, ideally, disproved. Such is the f undamental nature of science, as Platt long ago noted: “for exploring the unknown, there is no faster method” than the disproof of multiple alternative hypotheses.110 No other method yields scientific truth, much less correct mechanisms. The above crucial point about the disproof-based nature of both mechanistic models and science in general is, in our experience, lost on many dynamic model builders in science that do not have at least some training in rigorous, classical physical-organic mechanistic science and its guiding fundamental principle of Ockham’s razor.111 It follows that the power of good fits to the two-step sintering mechanism presented herein holds considerable potential to help uncover kinetically valid, disproof-based, more detailed, and more reliable atomistic sintering mechanisms that one can then trust and build from. In support of the above point, recently other mechanistic groups have been able to expand beyond initial two-step fits (i.e., in nonsintering areas) en route to deducing the more intimate mechanisms underlying, for example,77 Agn and78 Aum nanoparticle formation reactions, and in the photochemical dimerization of 9-methyl-anthracene in the solid-state for which attempts to use the empirical Avrami equation proved problematic.91 We highly recommend these seminal papers77−79,82,91 to interested readers, especially those wishing to understand better how fits to the two-pseudoelementary steps of the two-step mechanism hold insights into the underlying, elementary-steps mechanism that one is striving to uncover.

mechanism, implications which merit further testing and support or disproof, are (i) that avoiding syntheses and processes that form particles of different sizes will reduce sintering (because the kinetically dominant step between smaller and larger nanoparticles, B + C → 1.5C, will be slowed); (ii) that sintering will be slowed by site-isolation of nanoparticle catalysts (as is of course exceedingly well-known and hence currently commonly done, by anchoring them to solid supports), while also (iii) keeping the nanoparticles as site-isolated and “dilute” on the support as possible, so that the bimolecular nucleation of sintering, B + B → C, is inhibited (α 1/[B0]2) so that, in turn, the significantly faster, autocatalytic sintering process cannot start. Additional predictions are (iv) that the available activation parameters for the k3 and k4 steps (in solution−nanoparticle systems103) suggests that lower temperatures will avoid turning on the (higher ΔH‡) k4 autocatalytic agglomeration step, sintering being well-known to be highly temperature sensitive.44−47 In addition and more speculatively is the hypothesis (v) that added substrates that serve as nanoparticle ligands may well also be able to turn off the two-step mechanism64,103 and, possibly, turn on OR (vide inf ra). While there are seminal studies of the particle-size dependence of sintering that demand citation here,14,15 the mechanism-based insights (i), (iii), and (iv) did not have the kinetic and disproof-based mechanistic support that they now have. This is fully understandable and as expected because the discovery that the two-step mechanism provides excellent fits to such literature sintering data had not been reported until now. Future Possible Exploitation of the Initial Two-Step Curve-Fits En Route to the More Intimate, More Elementary-Step Based Catalyst Sintering Mechanism. Potentially quite significant for future studies, but presently little appreciated, is that when excellent curve-fits to the twostep kinetic model can be obtained and make physical sense, then that initial kinetic success opens up a previously unavailable, rational, increasingly precedented pathway77−79,82 for discovering the previously hidden, true elementary-step/ desired atomistic reactions that sum to the essential kinetic features (vide inf ra) of the two-pseudoelementary-step model used for the curve-fit.77−79,82 Put another way, good fits to pseudoelementary steps such as autocatalysis, A + B → 2B or B + C → 1.5C, are now known to contain kinetic information that more atomistic mechanisms must be able to reproduce or if they cannot, those postulated mechanisms are likely incorrect.77−79,82 One especially illustrative specific example of the above claim82 is presented in some detail in the Supporting Information for readers doubting this claim. Hence, initial fits to the two- (and four-) pseudoelementary step mechanisms can and should be used to support, or disprove, otherwise very reasonable, physical chemical/atomistic mechanisms that are physical-law based, seem to be highly reasonable overall, and are intellectually attractive chemically in that they are atomistic. However, many such kinetics models in the end analysis fulfill only the necessary condition (i.e., they can account for the data), but are not suf f icient to be the correct, true mechanismsince they lack a sufficient disproof basis. That is, they could be correct, but most often an experienced mechanistic chemist’s experience and hence hypothesis is that such proposed models are, actually, not correctclose, maybe, and often containing additional atomistic details well-supported by other physical methods, typically, but in the end analysis still not sufficient to be as reliable as extensively disproof-based mechanisms for which all other reasonable mechanismsall

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SUMMARY AND CONCLUSIONS Herein we have demonstrated for the first time the ability of the minimal two-step, B + B → C, B + C → 1.5C mechanism to fit sintering kinetics data previously accounted for only by empirical power-law fits. The balanced reactions of the twostep mechanism define both the rate constants k3 and k4 and, crucially, also the proper words needed to describe the process correctly physicallynamely bimolecular nucleation of agglomeration and then autocatalytic agglomeration. These two, balanced-equation-defined processes can now be used to fit, and where good fits are obtained, to describe the kinetics of supported-nanocluster catalyst sintering previously described only and nonmechanistically by empirical power-law treatments. We also have examined other literature catalyst sintering systems, the results of which support the hypothesis that the B + B → C, B + C → 1.5C mechanism is both physically reasonable and consistent with recent sintering data for different size nanoparticles. As such, a hypothesis going forward has to be that the two-step sintering mechanism is expected to prove more generally applicable to catalyst sintering of supported nanoparticle and related systems. We have provided other evidence that supports the novel finding herein that sintering should be viewed as a nucleated, autocatalytic processwhich in turn provides the insight that if bimolecular nucleation can be inhibited, then sintering should be greatly slowed. We have mathematically equated (as part of the Supporting Information materials) eq 2 and the power-law eq 6 (since these equations provide statistically equivalent fits) with the finding that the phenomenological constant “A” of the power laws, presented in the literature as an “Arrhenius parameter”,12 is better viewed as a convolution of the nucleation and K


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Industrial & Engineering Chemistry Research autocatalytic agglomeration rate constants k3 and k4 along with the concentration term, [B]0. Calling the adjustable “A” parameter of the power laws an “Arrhenius parameter” as done in the literature is, in our view, a classic example of the physically highly misleading conf usion that invariably results when one tries (in vain) to find physically relevant words that correspond to the adjustable, empirical parameters of such nonmechanistic, empirical parameter rate equations. Additionally, we have therefore shown herein that an extremely important feature of the minimal two-step, B + B → C, B + C → 1.5C sintering mechanism is its physically descriptive, balanced-reaction-defined concepts and associated physically meaningful words for describing the sintering kinetics data: bimolecular nucleation of agglomeration, followed by autocatalytic agglomeration. We have demonstrated that, for each set of data fit to the B + B → C, B + C → 1.5C mechanism, the autocatalytic rate constant k4 is greater than the nucleation rate constant k3and larger by up to a factor of 105 in some instances. This suggests the important insight that only a small amount of B + B → C nucleation of sintering is needed to “turn on” the faster, autocatalytic, B + C → 1.5C step. This is another significant, previously unavailable insight compared to all of the prior kinetic and mechanistic literature of sintering, to our knowledge. We have also discussed what is known about why the larger particles C appear to be more reactive than the smaller ones B; C particles appear to be more coordinatively unsaturated, which in turn implies that the type and concentrations of added ligands are predicted (i.e., hypothesized) to have important effects on catalyst sintering. We have detailed the hypotheses which result that demand further testing and support or refutation, including the hypotheses that larger, C, particles enhance sintering of supported-nanoparticle catalysts since they have on-average lower average metal−ligand (i.e., including M-support) bond dissociation energies,64 and that average larger C nanoparticles may be more mobile under nonadded ligand conditions compared to smaller B particles, at least for systems first shown to obey the two-step mechanism in Scheme 2. Support for, or refutation of, the above hypotheses will require additional experimental scrutiny by the community under a range of carefully selected experimental conditions that include different ligands, the metal/ligand ratio, temperature, and the presence or absence of any solvent or other relevant variables. In addition, we have provided the hypothesis for future research that two-step bimolecular nucleation and then the autocatalytic agglomeration sintering mechanism may be the best present description of the minimal chemical mechanism underlying “Particle Migration and Coalescence”, one of the dominant (previously phenomenological) descriptions of nanoparticle catalyst sintering. Both of the steps B + B → C and B + C → 1.5C do physically involve “particle migration and coalescence”, but these two pseudoelementary steps64 go beyond the word-only, qualitative descriptions of “PMC” and define precisely and chemically the associated rate constants and the associated concepts and proper words for describing physically what is occurring, PMC apparently equaling “bimolecular nucleation of agglomeration plus autocatalytic agglomeration”. We provide preliminary evidence that the “two-step PMC” is distinct from at least the best fitting, “three-step OR” and from the other two, deliberately minimalistic OR mechanisms

considered as part of this work. However, additional work on a minimalistic OR mechanism is needed, and continuing. We conclude that, on the basis of the two-step sintering mechanism, efforts to prevent nanoparticle sintering should focus on inhibition of the nucleation step, including avoiding syntheses that lead to particles of different sizes, site-isolating the nanoparticle on the support (as is well-known and hence commonly done1−12), and reducing the effective concentration of the nanoparticles on the support, with the acknowledgment again that a number of these are apparent from the extant sintering literature16,112,113although they did not have the firmer kinetic and mechanistic basis that they now have. Other mechanism-based control of sintering may be possible by using the temperature dependence of the k3 vs k4 agglomerative sintering steps to control sintering, and using the effects of added ligands64,103 to control sintering since the k3 step looks to be associatively activated, while the k4 step appears to be dissociatively activated.103 These approaches (varying temperature and ligands) are not new by any means, but such studies can now be designed and then analyzed from a firmer mechanistic basis. Finally, some important caveats to the use of the two-step nucleation and autocatalytic sintering mechanism were touched on earlier,68,90 but merit re-emphasis (see also the section elsewhere entitled “limitations...”64). • The two-step sintering mechanism is once again, by design, a minimal mechanism that obeys Ockham’s razor by deliberately being as simple as possible. Its pseudoelementary step kinetic treatment therefore hides by averaging many important kinetic and hence chemical/atomistic detailsnotably all the details of the underlying, truly elementary steps. For example, the dependence of the rate constants on the size of the particle are hidden in the average rate constants, k3 and k4. Also hidden in the model and the average k3 and k4 rate constants is the likelihood that the number of surface atoms in a given nanoparticle B or C (surface atoms treatable in an average or more continuous way114) are presumably more important than the total number of atoms present. Any and all metastable, kinetically important intermediates (such as possibly more stable, “magic-number”, nanoparticles) are also hidden within the two averaged species B and C. • Additionally, ill-understood but probable effects of the breadth of the particle-size distribution are not taken into account by the two-step or four-step models, despite the general prediction discussed above that broader size distributions are predicted to have higher sintering rates. In this regard, the reader is directed to seminal efforts in the literature to determine the effects of nanoparticle size/size-dispersity on sintering.15,16,106 • Interesting effects on, for example, the k4/k3 ratio are already apparent but remain poorly explained and, hence, would appear to be fertile ground for additional, needed studies. A good example is the Pdn/Al2O3 system examined in the Supporting Information for which the resultant curve-fit k4/k3 increases from 39 to 52 to 154 with a corresponding increasing in the %Pd loading from 1.0%, to 3.6%, and to 7.0%. See the Supporting Information for additional details and discussion of a list of possible factors that might begin to explain those interesting observations. L


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Industrial & Engineering Chemistry Research • Finally, OR is believed to operate at least in part in many sintering studies.59−63 While we have presented herein preliminary attempts to find a minimal OR kinetics model, and while those preliminary results appear to show that the “two-step PMC” model differs from OR, whether or not the current two-step mechanism can unequivocally distinguish PMC from OR is still an open question in our view, as discussed more in the Supporting Information for the interested reader. Hence, further kinetics and other testing of systems such as those cleverly employed in the 2014 study106 of monomodal Pt68 nanoparticles, in comparison to a deliberate mixture of Pt68 and Pt22 particles, seem warranted to try to definitively distinguish OR from PMC. One speculative hypothesis here is that the presence of suitable ligands that should accelerate a “Bn + mL → Bn‑1 + B1Lm” process (and perhaps also inhibit B + C → 1.5C type steps64,103), could be one key to favoring OR over PMC. The ideal end result still needed is a kinetics scheme that can easily be used, and hence will routinely be used, to curve-fit and distinguish OR from PMC. However, and overall, the results presented herein argue strongly that the two-step bimolecular nucleation and then autocatalytic agglomeration sintering mechanism presented in Scheme 2, and used herein to analyze sintering kinetics data previously fit by just empirical power-laws, is an advance toward a better, balanced-equations-based, hence more rigorous, disproof-based-mechanistic understanding of catalyst sintering. That said, we emphasize once more that the two-step sintering model is deliberately just a minimalistic, beginning foundation for the needed additional studies and development of more complex, more realistic, more atomistic mechanistic models of the complex phenomenon of particle agglomeration115 and sintering. More advanced sintering models built off a foundation of the minimalistic two-step model will need to account for known variables the two-step model presently neglects, including surface area, surface defects, surface ligands/ poisons, their dissociation constants, surface heterogeneity,108 and so on. Melding the existing, valuable atomistic, physicalchemical models with the present minimalistic kinetic model is an important, future goal. Hence, much work remains to be done to meld the trifurcation of knowledge in the sintering area documented in the Introduction as well as to further test, verify, and expand, or refute, the conclusions and hypotheses that have resulted from the present work. The overall goal continues to be the mechanistically guided preparation and maintenance of highly active, long-lasting, sintering resistant, supported- and other-nanoparticle catalysts.

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step mechanisms; verification that the value of [B]0 used in eq 2 affects the absolute values of k3 and k4, but not their ratio; mathematically equating eqs 2 and 6 in search of any resultant insights; two additional literature systems that demonstrate the generality of the two-step mechanism; two additional alternative Ostwald ripening mechanisms and their attempted fits to the data; discussion of the complexities of agglomeration phenomenon (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: fi[email protected]. *E-mail: [email protected]. ORCID

Eric E. Finney: 0000-0003-1793-8866 Richard G. Finke: 0000-0002-3668-7903 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported at Colorado State University by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences & Biosciences, vial DOE Grant SE-FG40203ER15453.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b02633. Prior use of the two-Step FW mechanism to account for solid-state kinetics data previously accounted for primarily by empirical or semiempirical equations; an illustrative example of where a fit to composite, pseudoelementary steps revealed underling insights that allowed the atomistic mechanisms to be discovered; a discussion of the use of atomistic models vs phenomenological equations; fitting with either the unimolecular or the bimolecular nucleation versions of the FW twoM


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