Gold Nanoparticle Formation Kinetics and Mechanism: A Critical Analysis of the "Redox Crystallization" Mechanism.

A 2013 paper proposed a "redox crystallization" (R-C) mechanism for the formation of Au0 n nanoparticles from the reduction of a AuCl4 - precursor. That study used an unconventional analysis of the valuable, expertly obtained kinetics data reported, and came up with multiple claims and insights collected under the putatively new R-C mechanism. If confirmed, those claims and the R-C mechanism provide a valuable addition to the knowledge of gold nanoparticle formation kinetics and mechanisms. On the other hand, if the methodology used to support the R-C mechanism is flawed so that its resultant conclusions are incorrect, then the R-C mechanism needs to be discarded until compelling evidence for it can be gathered, evidence that would have to include the disproof of the other dominant mechanism(s) of nanoparticle formation. The present work provides a critical analysis of the evidence previously offered for the R-C mechanism, efforts that are of interest to the areas of Au0 n nanoparticles, the kinetics and mechanisms of nanoparticle formation and, as it turns out, more generally to those interested in kinetic and mechanistic studies.

Introduction

Previously Proposed “Redox Crystallization” (R-C) Mechanism for Au0 Nanoparticle Formation

A 2013 paper presented valuable sigmoidal kinetics data (Figure a) for the gold nanoparticle formation system summarized in the caption of Figure . The date were analyzed in terms of a so-called “redox crystallization” (R-C) mechanism, shown in Scheme .[1]

Figure 1

Formation of gold nanoparticles from an aqueous solution of HAuCl4 in the presence of l-ascorbic acid (AA) and polyvinylpyrrolidone (PVP).[1] (a) Time evolution of the absorbance at 526 nm (surface plasmon resonance peak of gold nanoparticles). (b) Time evolution of the absorbance at 526 nm, normalized to the absorbance maximum in (a), labeled “zeroth-order” kinetics. (c) Time evolution of ln(1/(1 – x)), where x represents the normalized absorbance in (b), labeled “first-order” kinetics. (d) Time evolution of (1/(1 – x)) – 1, where x represents the normalized absorbance in (b), labeled “second-order” kinetics. Reproduced with permission from ref (1). Copyright 2013 Elsevier.

Scheme 1

Previously Proposed Redox Crystallization Mechanism of Gold Nanoparticle Formation

Adapted from eqs. F-1, F-2, and F-3 in ref (1). M represents gold precursor ions, M0l free soluble gold atoms, and M0s gold atoms in the solid state, whereas s* is surface active sites and R is the reductant (in the original article, R was shown as m in the equation but referred to as R in the text (and as used herein), the m in all probability being just a typographical error). Kc represents the redox equilibrium constant, and k01 and k02 are the nucleation and growth rate constants, respectively. In the proposed[1] R-C mechanism, step (a) corresponds to the reversible reduction of gold precursor ions into free soluble gold atoms, step (b) to the phase transformation of free soluble gold atoms into the solid state via nucleation, and step (c) to the growth of nuclei via interaction of free soluble gold atoms with surface active sites.

Under conditions where the authors defined the overall rate constant for nucleation as k1 = αk01 and for growth as k2 = αε[M]0k02, the proposed R-C mechanism yields the integrated kinetic eq 7 in the R-C paper,[1] reproduced as eq below. In eq , Abs represents the absorbance at time t for the surface plasmon resonance (SPR) peak of the gold nanoparticles, measured at 526 nm, and Absmax represents the maximum absorbance (it should be noted that a typo in eq 7 of the 2013 paper[1] was later corrected in eq 1 of a 2015 review[2]). The SPR absorbance data were used as recorded (i.e., without any corrections for particle size, shape, surface ligand changes, or other factors that can influence the plasmon resonance[3−6]). In the overall rate constant expressions for k1 and k2, the term α accounts for the proposed redox equilibrium, ; ε represents the fraction of surface “active” sites, ε = [3 × molar mass (Au)]/[density (Au) × average radius of Au nanoparticle], and k01 and k02 are the rate constants for nucleation and growth, respectively (see full derivation of eq in the Supporting Information, part A).A number of aspects and claims of the R-C mechanism paper (vide infra) quickly catch the interest of anyone interested in Au0 nanoparticles, in nanoparticle formation mechanisms, and in proper kinetics and mechanism in general.[1] Two aspects of the paper[1] that immediately caught our eyes are the statements that “Interestingly, (the) R-C model has (an) identical mathematical expression as the Finke-Watzky model...”[1] (hereafter the FW 2-step model), yet the conflicting statement in the Supporting Information claiming that the F–W mechanism “...is totally different from the R-C mechanism”.[1] Additionally, a very unconventional kinetic treatment was used to provide evidence for the R-C mechanism, and the overall approach to elucidating the mechanism was to try to “verify the applicability and validate the mechanism”[1] rather than the proper scientific approach of attempted disproof of all possible mechanisms.[7,8] It is that totally different mechanism and other claims, the unconventional approach to the kinetics, plus the lack of attempted disproof of other mechanisms of nanoparticle formation that are the focus of the present contribution.

Well-Precedented Alternative Mechanism of Nanoparticle Formation: The 1997 FW 2-Step Mechanism of Slow, Continuous Nucleation and Autocatalytic Surface Growth

The currently most highly cited (>500 citations as of 2017, according to SciFinder), accepted minimal mechanism for nanoparticle “slow continuous nucleation and fast autocatalytic surface growth” is the FW 2-step model, shown in Scheme , with its associated integrated kinetic eq .[9] Of note here is that the FW 2-step mechanism has been shown to account for gold nanoparticle formation by other research groups.[10−12]

Scheme 2

FW 2-Step Mechanism of Slow Nucleation and Fast Autocatalytic Surface Growth[9] Shown in Its Simplest and Then Generalized Form[13,14]

Here, A represents precursor metal ions (e.g., nominally AuCl4– in the present example) and B the growing metal nanocluster past its nucleus size (rigorously, by stoichiometry B = Au0 in the present example), whereas k1 and k2 are the average rate constants for nucleation and growth, steps (a) and (b), respectively.

The integrated kinetic eq shows the concentration of B, the growing metal nanocluster, expressed as a function of time, the average rate constants for nucleation and growth, k1 and k2, respectively, and [A]0, the initial concentration of metal precursor ions A.[9]To account for surface sites involved in autocatalytic surface growth (i.e., for the B = Au0 that are covered up in the growing nanoparticle), a scaling factor was originally developed in 1997 and applied to include B atoms on the surface of the growing nanocluster only (see footnote 46 and Appendix C in ref (9)), that is, xgrowth = increase in the number of surface atoms/increase in the total numbers of atoms, with the scaling correction factor applied to the value of the growth rate constant extracted from a curve fit to eq , . Since then, Schmidt and Smirnov have developed a continuous function based on the FW model that describes the dependence of the number of surface atoms on the total number of atoms.[15]

Also relevant to what follows is that, as apparent rate constants for (composite) pseudoelementary steps, k1 and k2 are expected to show concentration dependence(s). Indeed, in the original 1997 FW paper,[9] both k1 and k2 showed a dependence on the reductant H2 concentration; see Figures 5 and 7 and footnote 37 in ref (9).

Questions Raised by an Initial Comparison of the FW 2-Step and R-C Mechanisms

It is easily seen that the overall rate constants proposed by the R-C model coincide conceptually with the pseudoelementary rate constants defined by FW (eq ) and recalling the terms defined above:A comparison of the integrated kinetic eqs and 2 shows that, indeed and as the R-C mechanism paper notes, these two equations are identical mathematically. It follows that the basic mechanism underlying both the FW 2-step mechanism and the “R-C” mechanism must also be very close, if not functionally identical. However, as already briefly noted in their Supporting Information, the authors of the 2013 paper claim that the FW mechanism “...is totally different from the R-C mechanism”.[1] Moreover, the R-C model has since reappeared in a 2015 paper,[2] where it was presented as novel and without reference to the 1997 FW 2-step mechanism, that is, as if the R-C model is indeed as claimed “totally different”[1] from the FW 2-step mechanism. Is this “totally different” claim actually supported? If totally different, it would be an important advance in the mechanisms of nanoparticle nucleation and growth. Indeed, a totally different mechanism hidden under identical mathematics would, if correct, be of broad importance to both kinetics and mechanisms, in general, and specifically to nucleation and growth across nature given the broad range of sigmoidal curves accounted for at present by the FW 2-step mechanism.[10,13,14,16−26]

Hence, the postulated R-C model raises several questions. (i) Is the R-C model actually novel and different from the 1997 FW 2-step mechanism as claimed?[1,2] Second, (ii) does it make sense kinetically and mathematically to do what the authors of the 2013 paper did, in their unconventional treatment of the kinetics data, to try to obtain evidence for the R-C mechanism? Specifically, can one as they did plot sigmoidal kinetic data by separate zeroth-, first-, and second-order plots, as reproduced, respectively, in Figure 1b–d from the 2013 paper,[1] and then “cherry pick” each small, linear part of just a section of those separate plots and then also interpret that small linear section as evidence for mechanistic steps that are zeroth-, first-, and then second-order throughout the whole kinetic curve? Can one do this even though the linear regions are observed for only brief time periods of the full kinetic curve? Does this unconventional kinetic treatment, which differs from Michaelis–Menten enzyme saturation kinetics[27] or graphical rate equations[28] and does not provide the normal “sufficient high quality [kinetic] data over the course of the reaction”,[28] make sense? (The authors themselves note that “Such sigmoidal kinetic curves were for the first time analyzed by conventional rate laws”.[1]) (iii) Crucially, a fit to the well-precedented FW 2-step mechanism was not tried previously but should have been the first attempt to fit the data in the 2013 paper. In the end analysis then, (iv) is the R-C mechanism something new and novel, with value to the nanoparticle nucleation, growth, and agglomeration kinetic and mechanistic literature, or not? It is the answers to these four primary questions that are the focus of the present study, a critical reanalysis of the 2013 paper that resulted in the redox crystallization mechanism. The results should be of interest to anyone interested in Au0 nanoparticles, in the mechanisms of nanoparticle formation, and in proper kinetic and mechanistic analyses, in general.

Results and Discussion

FW 2-Step Mechanism Fits the Data in the 2013 Paper

Figure shows a fit of the FW 2-step mechanism and associated integrated kinetics equation (eq above) to the Figure 1b kinetics data in the 2013 paper.[1] The fit is excellent according to the R2 = 0.9987 and by visual examination. The SPR absorbance data were used without correction for possible size, shape, ligand, or other effects[3−6] as done in the 2013 paper.[1] The fit to the FW 2-step mechanism is excellent[29] and suggests that any effects on the surface plasmon resonance (when expressed per B = Au0 atom in the growing nanocluster) are either relatively small over the range of nucleation and growth monitored and/or convoluted into one or more of the measured rate constants (if so, then mostly into k2 is what one would expect since the kinetically effective nucleus[30,31] is likely small, just a couple of atoms being precedented for Ag nanos[32,33]). The former explanation is consistent with recent studies from one of our laboratories (MW) that found similar rate constants within experimental error for Ag(0) nanoparticle formation followed by the plasmon resonance compared to two different, more direct methods.[34] That said, there are a number of issues to consider when using the SPR to monitor nanoparticle formation kinetics, a discussion of which is provided in the Supporting Information.

Figure 2

Product fraction data (x = Abs/Absmax) digitized from Figure b[1] and curve-fitted to the FW 2-step mechanism.[9] Curve fit values are k1 = 2.19 × 10–5 s–1, k2 = 0.234 s–1 for [A]0 = 1.00, with R2 = 0.9987.

The excellent fit to the FW 2-step mechanism is expected because this mechanism has been shown to account for a very wide range of nanoparticle formation systems under reductive and other conditions,[35−38] including Au0 nanoparticles.[10−12,29] Hence, just from the fit in Figure to the FW 2-step mechanism, by literature precedent,[10,29,35−38] and with the application of Ockham’s razor (i.e., until there is additional compelling evidence demanding a more complicated or different mechanism), the FW 2-step mechanism is the best currently precedented way to minimally, but efficiently and effectively, describe the system in Scheme and kinetics data in Figure b. This was not done before as part of the 2013 work.[1]

Re-examination of the Use in the 2013 Paper of Separate Linear Zeroth-, First-, and Second-Order Plots, of Selected Sections of the Kinetics Curve, To Offer Putative Evidence for the Reaction Mechanism over the Full Time Course of the Reaction, Including for the Prior Equilibrium Step Proposed in the R-C Mechanism

A look back at the R-C mechanism in Scheme 1 from the 2013 paper, and in comparison to the FW 2-step mechanism in Scheme , reveals that the only true difference (vide infra, section ) is the reductive prior equilibrium of gold precursor ions, M in Scheme , by the reductant, R (l-ascorbic acid in Figure data), to putatively produce soluble gold atoms (M01), that is, M + R ⇆ M0l (equilibrium constant Kc), as postulated in the R-C mechanism (Scheme ). Is there really an equilibrium, as opposed to the dependence on reductant already established in 1997 for the FW 2-step mechanism (H2 in that case[9])? Hence, it follows that the next crucial task is to examine carefully and critically the evidence in the 2013 paper for the addition of the prior equilibrium, M+ R ⇆ M0l.

In their analysis of the kinetics data, the authors stated that[1] “the reduction of the gold precursors was relatively faster than the crystallization process, i.e., the latter... becomes the rate-determining step during... [gold nanoparticle] formation” and “Therefore..., we assume reduction of gold precursor (M) by reductants (R) as a reversible reaction... with an equilibrium constant Kc.” Restated, in their R-C mechanism, the authors set up a faster “reduction” first step (step (a) in Scheme ) that then acts as a bottleneck for the slower “crystallization” second step (nucleation and growth, steps (b) and (c) in Scheme ). Hence, their interpretation of the kinetics data, and its use as “supporting evidence” for the above, is what needs to be examined in some detail next.

The authors expressed a product (or “crystallization”) fraction x corresponding to the normalized absorbance at 526 nm for the SPR peak of the gold nanoparticles.[1] A plot of x versus time produced a “zeroth-order” graph (Figure b, vide supra, for zeroth-order kinetics “rate = constant”) that exhibits a linear region from just ca. 34–48 s (or 8 data points).[1] The authors claimed from this that “the overall crystallization rate was zeroth order with respect to the reactants (namely gold atoms in this case)” because “excessive gold monomers competed for relatively limited active sites” and that “at this stage all of the Au(III) species had either been reduced to its atomic state or into its lower charge states”. A plot of ln(1/(1 – x)) versus time produced a “first-order” graph (Figure c, vide supra, for first-order kinetics, rate = constant × reactant fraction) that exhibits a linear region from just ca. 46–54 s (or 5 data points).[1] A plot of (1/(1 – x)) – 1 versus time produced a “second-order” graph (Figure d, vide supra, for second-order kinetics, rate = constant × (reactant fraction)2) that exhibits a linear region from just ca. 52–56 s (or 3 data points).[1] The authors concluded that, “with the crystallization proceeding forward, it changed to first order... and subsequently to second order... which indicated that the concentration of gold monomers declined and its influence became evident on the crystallization rate” and that it “resembled... Michaelis-Menten kinetics in enzyme catalysis”.

Here, the authors refer to enzyme saturation kinetics,[27] in which a zeroth-order dependence of the reaction rate on substrate concentration is expected at larger substrate concentrations, with a first-order dependence expected at lower concentrations. Hence, and as a test of these claims, herein we employed graphical rate equations (reaction progress kinetic analysis[28]) as a way to help visualize the reaction rate dependence on reactant concentration (or fraction of reaction). Specifically, for the data reported in Figure b, the reaction rate was calculated from the graph and expressed as a function of time (Figure ) or as a function of the reactant fraction, 1 – x (Figure ; a similar plot could be expressed as a function of the product fraction x). Note that the reaction progress in Figure should be read from right to left. The highlighted data points corresponding to the 34 to 48 s time period show that the reaction rate is not constant there (Figure ) nor does it display a zeroth order (“flat”) dependence on reactant concentration (Figure ). The fact that a linear portion of the growth curve (Figure b) exists around its inflection point is explained in section and, in fact, is predicted by the FW mechanism (vide infra). Similar graphical rate equations[28] were constructed to look at a first- or second-order dependence of the reaction rate on reactant concentration (fraction) and are shown in part B of the Supporting Information.

Figure 3

Reaction rate (dx/dt, with x representing the product fraction) calculated graphically from the data in Figure b[1] and plotted versus time. Data points corresponding to the time period of 34–48 s (see Figure b) are highlighted. The key point to note is that the reaction rate is not constant over this interval, which in turn invalidates any claim of a constant, zeroth-order reaction over this interval.

Figure 4

Reaction rate (dx/dt, with x representing the product fraction) calculated graphically from the data in Figure b[1] and plotted versus the reactant fraction 1 – x. Here, reaction progress should be read from right to left (1 – x = 1 to 1 – x = 0).[28] Data points corresponding to the time period of 34 to 48 s (see Figure b, vide supra) are highlighted. The take-home message is again that the reaction rate is not constant; hence, the reaction is not the previously claimed zeroth order[1] over this interval.

Beyond the disproof of the zeroth-order claim provided by Figures and 4, reflection reveals that assuming a reaction order, from any plot that shows that order for just a fraction of the whole reaction, is not a valid kinetics approach. The above reinforces the well-known rule of chemical kinetics: the integrated rate equation must fit all of the reaction time course under consideration. This is a fundamental rule of chemical kinetics that one cannot violate. The same analysis as above also disproves the use of first-order and second-order plots in the 2013 paper,[1] as reproduced in Figure c,d, vide supra.

Note that we do not mean to imply here that biphasic, triphasic, or even higher-phasic kinetic curves are not seen in chemical science. In those cases, yes, one is forced to use more than one kinetic fit, but even then only for different (but whole; complete) portions of the kinetic curve and only after (i) showing that no single mechanism is able to fit the whole curve, and even then, (ii) the postulated second, say biphasic part of the kinetics must, itself, be well-fit by that second proposed mechanism over the whole second part of the (second half) of the kinetic curve. Highly relevant to the present case and the R-C model, then, is that a single mechanism and its associated integrated rate equation is able to fit the full kinetic data offered in the 2013 paper, namely, the FW 2-step mechanism, as shown in Figure .

Understanding the Origin of the Linear Part of Figure b

It is of some interest at this point to examine what the linear part of Figure b actually corresponds to and for the sake of a better understanding of sigmoidal curves so common in nucleation and growth phase change processes across nature. Recently, we took the first, second, and third derivatives (the latter the so-called “jerk”) of our eq , corresponding to the FW 2-step mechanism, to learn more about what controls the shape and key points such as the induction period, linear section, and inflection point of such sigmoidal growth curves.[39] While we refer the interested reader to that work, the most relevant part for the present paper is what the linear portion of the kinetic curve in Figure b corresponds to (eq and also in Figure a, if concentration is expressed instead of fraction):As eq teaches, the linear portion or maximum slope is a function of k1, k2, and [A]0, not time; it is thus constant or “zeroth order” in that sense. Indeed, the maximum slope is found around the inflection point of the sigmoidal curve, at a time where the rate reaches a maximum and the acceleration changes sign.[39] The FW 2-step mechanism further teaches why, phenomenologically, a linear portion in the sigmoidal curve is seen: the rate of nucleation plus its magnification by autocatalytic growth has reached a maximum, and the depleted concentration of [A] means that the reaction is slowing down, overall exhibiting a composite region that just happens to be linear and which obeys eq . The analytic expressions for the tinduction, tinflection-point, and [B]max at the inflection point are also available elsewhere for the interested reader.[39]

R-C Mechanism Does Not Yield a Different Integrated Rate Equation, Does Not Yield Any Evidence for the Proposed Prior Equilibrium, and Does Not Yield a Value for Its Putative Kc Equilibrium Constant

An important practical point can be raised at this juncture: as is obvious by a comparison of eq for the R-C mechanism to eq for the FW 2-step mechanism, as well as by the author’s own admission that the “R-C model has (an) identical mathematical expression as the Finke-Watzky model...”,[1] fits to either the R-C model or the FW 2-step mechanism yield identical phenomenological observables, k1and k2in the FW 2-step model nomenclature, and with k2(R-C) (=k2(FW)[A]0) in the R-C model. That is, practically speaking, even if one accepted the now disproven R-C mechanism, no new information results from its current implementation (i.e., no Kc value example for the putative prior equilibrium of the R-C model). The claimed “...totally different...” R-C model is, actually, mathematically as well as functionally identical to the FW 2-step model. The R-C model is, therefore, not new and should be abandoned until and unless compelling evidence for it is provided.

Worth noting here is that such “identical measurable rate constant results” will be the case when any, and any number, of fast Keq are added prior to the two steps of the FW mechanism (as the R-C mechanism does with its putative prior Kc). It is easy to show mathematically that adding prior fast Keq will just produce pseudoelementary steps that will have composite rate constants that are indistinguishable from k1 and k2 of the FW 2-step mechanism (see elsewhere for more on pseudoelementary steps[9]). Part A of the Supporting Information, which provides more detail on the derivation of eqs 1–7 in the 2013 R−C model paper,[1] also serves as a short mathematical proof of this point.

A reviewer (whom we thank) noted that one might use the term “irreducible” to describe the FW 2-step model (vs “reducible” for the postulated R−C mechanism). However, the term irreducible has a strict mathematical meaning that we do not wish to claim in the absence of the required, detailed, mathematical analysis. Rather, the FW 2-step model is based on a minimalistic “Ockham’s razor”[40] approach and yields pseudoelementary average rate constants which can sometimes be experimentally deconvoluted in favorable systems (vide infra).

Fundamental Underlying Problem with the Science in the 2013 Paper Is That They Attempted the—Impossible!—Proof of Their Proposed R-C Mechanism, Rather than Trying To Disprove It

It can been seen at this stage that the most fundamental, and hence greatest, underlying problem with the prior work leading to the R-C mechanism is the attempt therein to gather evidence to try to “prove” their postulated R-C mechanism. The instead needed correct approach of epistemologically sound science[7,8] is of course the disproof of all reasonable, multiple alternative hypothesis—in this case, all reasonable alternative mechanisms—followed by the application of Ockham’s razor[40] to choose the simplest mechanism consistent with the available data.

Notable in this regard are the now over 21 alternative mechanisms that have been disproven[41−44] en route to the highly disproof-based FW 2-step mechanism and its expanded 4-step mechanism of slow, continuous nucleation, autocatalytic surface growth, and two kinds of agglomeration that has been built off of (and hence which contain) the FW 2-step mechanism.[9,31,41−44]

Additional Issues with the Approach Used To Obtain and the Resultant R-C Mechanism

For the sake of completeness, the additional problems in the prior kinetic analysis and proposed R-C mechanism are dealt with in part C of the Supporting Information for the interested reader.

Despite the Current Broad Applicability of the Deliberately Minimalistic FW 2-Step Mechanism, It Is Important To Find Valid Examples Where the 2-Step Mechanism Can Be Expanded To Reveal the Always Present, Additional Steps Underlying the Pseudoelementary Step Treatment and the More Intimate, More Complex Mechanisms

The FW 2-step mechanism currently enjoys very broad, expanding application in many parts of the natural world, including to multiple metal nanoparticle formation systems[35−38]—including gold nanoparticles,[10] homogeneous catalyst formation,[16−18] heterogeneous catalyst formation,[19] catalyst sintering phenomena,[20] protein aggregation,[13,14,21,22] solid-state kinetics,[23,24] dye aggregation,[25] and other areas of nature.[26] The strengths of the 2-step mechanism and multiple insights it allows are summarized elsewhere for the interested reader, as are the weaknesses and caveats of the deliberately minimalistic, deliberately Ockham’s-razor-obeying, FW 2-step mechanism[44]—caveats and weaknesses that anyone using our mechanism needs to understand in order to use it properly. The weaknesses of the FW 2-step mechanism derive from its deliberately minimalistic, in the end analysis oversimplified, nature that thereby yields average results for systems that may contain 100s (to 1000s) of elementary steps. It follows, therefore, that a very important future effort is to expand upon and discover the more intimate, more complex mechanisms for nucleation, growth, and agglomeration systems that are well-fit initially by the FW 2-step mechanism.

There are currently five examples to our knowledge where systems that have been initially fit by the 2-step mechanism have been expanded to reveal their underlying, more complex mechanisms[10,16,24,45,46]—so that the needed work of the true, underlying mechanisms for sigmoidal curves fit by the FW 2-step mechanism is just starting and needs to be continued. (i) The first of these is our collaboration with the Bergman research group, where the development of a homogeneous catalyst showed sigmoidal, autocatalytic kinetics but where the true catalyst and the underlying catalytic mechanism remained completely obscure.[16] To get at the mechanism, a fit to the 2-step mechanism was done and yielded a good fit. That good fit, in turn, revealed that figuring out what “B” is was key, B being the true, but hidden, catalyst in the reaction. The catalyst B was then identified via trial and error writing of the multiple steps that had to add up to the A → B, then A + B → 2B pseudoelementary steps (see the highly illustrative Supporting Information provided with that paper[16]). This led to identification of the previously hidden PtCl(t-Bu3P)2(SnMe3) catalyst, after which it proved easy to write the catalytic cycle.[16] This example is the presently best available example we are aware of illustrating the power of the minimalistic, simple 2-step mechanism to reveal an underlying catalyst and more complex mechanism that, prior to the fit to the 2-step mechanism, was completely obscure and at an intractable standstill prior to the fit to the 2-step mechanism.[16]

A second example is (ii) by Kytsya and co-workers who, after a fit to the FW 2-step mechanism to their Ag+ plus N2H4 and NaOH, Ag0 nanoparticle formation system,[45] expertly expanded the kinetics to flush out the underlying reaction orders for all the reagents otherwise hidden by the two pseudoelementary steps in the minimalistic 2-step mechanism. A third example is (iii) by Sergievskaya and co-workers and is also for Ag+ reduction by N2H4, but now in the presence of reverse micelles. Here, these authors also elucidated the underlying, more detailed rate law and implied more intimate mechanism in greater detail than revealed by the initial fit to the 2-step mechanism.[46] Sergievskaya and co-workers (iv) also expanded greatly beyond the initial 2-step FW mechanism in a Au0 nanoparticle formation, again using N2H4 as the reductant, with a series of more detailed steps supported by additional kinetic evidence (steps and eqs 3–18 in their valuable, insightful paper[10]).

The final example is the very nice effort of Bardeen and co-workers who insightfully use, and especially then expand past, the FW 2-step model to uncover the more intimate mechanism of photochemical dimerization of 9-methylanthracene in the solid state.[24] We hope to see many more of these valuable mechanistic studies that use the FW 2-step mechanism as it is intended and best used—as a starting point for greater mechanistic insight into nucleated autocatalytic growth processes across in nature.

Conclusions

The primary conclusions from the present reanalysis of the “redox crystallization” model and the valuable, expertly collected data the authors provided therein can be summarized as follows:

The kinetic methodology used to try to support the R-C model, namely, finding small regions in sigmoidal curves that show linear sections for randomly selected (zeroth-, first-, and second-order) rate laws, is a conceptually and mathematically flawed approach that has no basis in modern chemical kinetics.

Reflection reveals the following: the precedented mechanism that will give the linear regions seen by the 2013 authors in their zeroth-, first-, and second-order plots—that is, the simple, minimal mechanism that those authors were trying to discover—is the FW 2-step mechanism. Put another way, the analytic function corresponding to the FW 2-step mechanism is the only currently known, minimal-mechanism-based function that will give the linear zeroth-, first-, and second-order plots that the authors see!

The putative R-C mechanism and its postulated prior Kc equilibrium have no experimental basis nor support at this time. It, too, should be discarded until and unless it receives compelling supporting evidence.

A comparison of the integrated rate equation for the R-C model to that established 20 years ago for the FW 2-step model (eq compared to eq , respectively) shows that they are identical mathematically and, hence, in the end will yield the (apparent, composite, pseudoelementary step rate constants) k1 and k2 of the FW 2-step model. Hence, the presently unsupported, herein disproven claim that the R-C model “...is totally different...” from the FW 2-step model needs to be abandoned until and unless compelling evidence to support such a claim is obtained.

A broader, important point is the demonstration herein that identical mathematical equations will result anytime fast prior Keq are added to the FW 2-step model since the proposed steps can be summed to pseudoelementary steps that still have primarily a A → B and then A + B → 2B form—see examples elsewhere that teach that even more complex equations and mechanisms can still show sigmoidal shape and be fit by the 2-step mechanism.[16,17] This means that efforts to expand beyond the FW 2-step model will need to provide compelling evidence for any additional equilibria or other proposed steps.

The above points so made, it is important to continue to strive to discover the underlying steps that are hidden in the disproof-based, deliberately minimalistic (so it can be as rigorous as possible), FW 2-step mechanism. Five examples where doing just that were presented,[10,16,24,45,46] along with references to the weaknesses and caveats as well as the strengths of the FW 2-step mechanism.[44] Those papers are recommended to anyone using the FW 2-step or using the more complex, but still minimalistic, disproof-based, and Ockham’s razor obeying 4-step mechanisms that are built off of the FW 2-step mechanism.[41−44]

Any future effort to understand the Au0 formation system in the 2013 paper, or any other nucleation, growth, or agglomeration system for that matter, will have to avoid the compounding mistake made in the 2013 paper, that is, the failure to read carefully and understand as fully as possible the huge, disparate, literature of nucleation, growth, and agglomeration processes across nature. It is becoming harder to discover anything new that has not already been proposed in a more general way in the broad literature of nucleation, growth, and agglomeration processes across nature, at least in our opinion. However, easier and still very important to discover are the more intimate, underlying mechanistic steps[10,16,24,45,46] that add up the FW 2-step kinetic scheme.

Finally, we end with the same important point we find ourselves ending many of our papers with these days: only a scientific method that consists of the attempted disproof of all reasonable, multiple alternative hypotheses is able to discover new mechanisms and, indeed, to verify new science.[7,8] As Platt has noted, “for exploring the unknown, there is no faster method” than the attempted disproof of multiple, alternative hypotheses.[8]

Methods

Data Digitization

Data were extracted and digitized from the published curves[1] using Digitizelt 2.2.1. As noted in the main text, the raw surface plasmon resonance absorbance data were used in the curve fitting.

Curve Fitting

The digitized data were fit to eq using OriginLab 2017 with a nonlinear curve-fitting function. The values of the rates constants k1 and k2 and of the initial precursor concentration [A]0 were obtained from the curve fit.

Graphs

Graphical differentiation was performed with OriginLab 2017. All graphs were plotted with Excel.