Kinetics and Thermodynamics of Efficient Chiral Symmetry Breaking

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Kinetics and Thermodynamics of Efficient Chiral Symmetry Breaking in Nearly Racemic Mixtures of Conglomerate Crystals Peter J. Skrdla 640 Maple Street, Westfield, New Jersey 07090, United States ABSTRACT: The remarkable evolution of homochirality in Viedma-type conversions, which typically start from racemic or nearly racemic mixtures of D and L conglomerate crystals, is examined using dispersive kinetic models. One such model for nucleation rate-limited phase transformations is found to describe well the sigmoidal (“S”-shaped) conversion transients of those processes better than both the classical first-order (simple exponential) mechanism proposed in the current literature and the JMAEK (“Avrami”) equation traditionally used to model nucleation-and-growth processes. That finding is discussed in light of the secondary nucleation observed in similar, Kondepudi-type conversions as well as the recently proposed mechanisms of attrition-enhanced Ostwald ripening and solution phase racemization.

’ INTRODUCTION The process recently discovered by Viedma,1,2 which can be referred to as “attrition-enhanced deracemization”, consistently provides the highest enantiomeric excess (ee) upgrades for compounds found to crystallize separately to form racemic D þ L conglomerates. While Uwaha3 was the first to attempt to describe Viedma's experiment in detail, Noorduin et al.4 first achieved a Viedma-type conversion with chiral molecules (as opposed to using achiral molecules that simply crystallize as racemic conglomerates). Such evolution of homochirality, recently demonstrated in the laboratory using biologically relevant molecules,5 might have potential implications for the origin of life on Earth (e.g. ref 6) and for the development of manufacturingscale processes for certain (e.g., pharmaceutical) compounds.7 However, due to the current ambiguity in the literature regarding the underlying mechanisms relevant to such conversions, this work draws on recent advances in dispersive kinetics theory8 to lend clarity to those mechanisms. Background. Some 160 years ago, Pasteur separated the enantiomorphic crystals of sodium ammonium tartrate using a pair of tweezers.9 While that was a fascinating discovery, such mechanical separation of conglomerates is not of much practical interest. Furthermore, the ability of achiral molecules to form chiral crystals is quite rare. Of greater interest is the separation of chiral molecules, in both the solution phase (e.g. ref 10) and in the solid-state (e.g. ref 11). With regard to the solid-state, this work is concerned with the ∼15% of known chiral molecules that crystallize to form racemic D þ L conglomerates5 (the majority form racemic compounds composed of both enantiomers in equal proportion12). In those cases, it has recently been demonstrated that one can forego chiral separation altogether, since it is r 2011 American Chemical Society

now possible to convert a racemic mixture of D þ L conglomerates into highly enantiopure crystals composed of molecules of either the D or L handedness, via a process called “chiral symmetry breaking”. Two decades ago, Kondepudi et al.13 first reported spontaneous chiral symmetry breaking in stirred, highly concentrated (supersaturated or “far-from-equilibrium”1315) solutions of the achiral compound, sodium chlorate, which crystallizes upon cooling to yield the chiral space group P2l3. To reproducibly obtain a majority of conglomerate crystals of only one handedness, those workers found that seeding with the desired crystal form was necessary (following Kipping and Pope16). Seeding, which imparts a slight excess of solids of one enantiomer, perturbs the system thermodynamics sufficiently to drive the evolution toward homochirality in a unidirectional manner. It does so by producing daughter crystals of the same optical rotation as the “mother crystal”, since the chiral symmetry breaking is rate-limited17 by a secondary nucleation process (vide infra). Since all symmetry-breaking events in nature are rare, as first recognized by Frank,18 in the absence of seeding, the Kondepudi process is initiated randomly, causing symmetry-breaking to evolve stochastically, i.e., toward either D or L conglomerates in equal proportion (provided one performs multiple experiments). Frank18 also hypothesized that once a small (but statistically significant) excess of one enantiomer in the system had been produced, in order to achieve chiral symmetry breaking a subsequent amplification mechanism, to replicate one Received: January 25, 2011 Revised: February 25, 2011 Published: March 16, 2011 1957

dx.doi.org/10.1021/cg200116e | Cryst. Growth Des. 2011, 11, 1957–1965

Crystal Growth & Design enantiomer while simultaneously destroying its mirror image, was needed. As mentioned above, the mechanism put forth by Kondepudi and co-workers1315 to explain their observations relied mainly on the process of secondary nucleation, which occurs following the mechanical disintegration of the seed (“Eve”19) crystal by the stir bar. The broken-up mother crystal provides nucleation sites for the rapid nucleation-and-growth of crystals with the same handedness as the seed, as videotaped by McBride and Carter.20 Unfortunately, as observed by Quian and Botsaris,21 the efficiency of the Kondepudi process can be quite variable, with lower supersaturations generally producing the highest enantioenrichment and vice versa. That is because high supersaturation favors primary nucleation over secondary nucleation, yet only the latter mechanism leads to enantioenrichment, since primary nucleation produces equal amounts of both enantiomer crystals, while only secondary nucleation produces nuclei that grow into crystals with the same handedness as the mother crystal. Therefore, control of the supersaturation (i.e., the system thermodynamics) and, consequently, the nucleation and/or growth rate (the kinetics) is critical to the process performance in terms of maximizing the ee of the final conglomerate crystal distribution. More specifically, since the GibbsThomson effect responsible, e.g., for particle coarsening or “Ostwald ripening (OR)22” favors the growth of larger crystals at the expense of smaller, less thermodynamically stable ones, there is increasingly less opportunity for crystals of the opposite-handedness to nucleate and grow once crystals of the desired chirality begin to form.23 That is, of course, unless there is a significant driving force with which to contend, such as that produced at high supersaturation, potentially allowing both primary and secondary nucleation to occur simultaneously. Therefore, in order to maximize ee, one should perform the NaClO3 symmetry-breaking experiment of Kondepudi and co-workers closer to equilibrium, i.e., with a lower driving force.24 Viedma1,2 was first to demonstrate the formation of enantiopure NaClO3 crystals, starting from systems of racemic mixtures of the two enantiomorphic solid phases in equilibrium with the achiral solution phase, via experiments performed under conditions of constant, lower (i.e., “closer-to-equilibrium”) thermodynamic driving force. He consistently achieved high ee upgrades for either of the two enantiomorphs, generated randomly (i.e., stochastically), by incorporating a crystal attrition mechanism effected by glass bead grinding. The extra mechanical energy input into the system helped produce continuous renewals of the crystal surfaces, both increasing the likelihood of secondary nucleation and facilitating crystal dissolution. Based on the success of Viedma using NaClO3 enantiomorphs, recent works have demonstrated the “glass bead grinding” approach to drive the emergence of a single solid chiral phase from a nearly racemic mixture of conglomerates of an amino acid derivative4 and of a proteinogenic amino acid,5 whereby each of the molecules of interest is also chiral in solution (unlike NaClO3) but able to interconvert (i.e., enantiomerize) as per Scheme 1. Recent works have touted the importance of an “attritionenhanced OR”4,5,25 mechanism and the “Meyerhoffer double solubility rule”26 in achieving chiral symmetry breaking in Viedma-type processes. For instance, it has been discussed that the solution racemization process (see Scheme 1), coined “chiral amnesia”,2,4 and/or the overall crystal surface area reduction caused by OR4,5,25 provide(s) the key driving force for the evolution of solid-phase single chirality. However, other works

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Scheme 1. Dissolved D and L Enantiomers of a Given Molecule Undergoing Solution-Phase Racemization, in Equilibrium with Solid Conglomerate Crystals of Each Chiral Species

have admitted the possibility that small, “sub-critical” clusters3,7,23 might also play a role in the evolution of homochirality, contributing to a lack of clarity with regard to the underlying mechanism. Along those lines, Noorduin et al.27 recently claimed to have found experimental evidence for the relevance of small clusters in the so-called “Viedma ripening” process, in accordance with the theoretical framework provided by Uwaha and Katsuno28 (discussed more later). Adding to the confusion, from the work of Frank18 and others (e.g. refs 1 and 29) it is known that the underlying mechanism should involve “nonlinear dynamics” (note that, e.g., solution phase autocatalysis typically produces sigmoidal, “S”-shaped kinetic profiles), yet the empirically observed kinetics for the evolution of system ee have recently been discussed in terms of a classical first-order mechanism7,25,26 (which relates logarithmic/ simple exponential profiles). As a result, it has been stated that “the detailed mechanistic understanding...continues to elude us”.26 To lend clarity to that issue, this work presents a new approach for modeling the solid-state kinetic profiles of Viedmatype processes, using dispersive kinetic models to provide fresh insight into the chiral symmetry breaking mechanism of Scheme 1.

’ RESULTS AND DISCUSSION Thermodynamics of Viedma-Type Processes. Plasson and Brandenburg24 recently discussed the need for energy influx into a given system in order to achieve chiral symmetry breaking, with the ultimate goal of total enantioenrichment. Without it, chiral molecules such as amino acids, whose enantiomers interconvert very slowly, would give rise to the “calamity of racemization”30 over long/geological time frames because that is the end state of the entropically favorable (spontaneous) process. Conversely, provided too much energy (i.e., a far-from-equilibrium condition), when rapid kinetics override the system thermodynamics, random primary nucleation can result in a negligible ee enhancement in the solid phase.29 Viedma-type (glass beadgrinding, closer to equilibrium) conversions, in which the energy input is controlled by both temperature and mixing conditions, appear to achieve a balance between the underlying kinetics and thermodynamics2 required for efficient chiral symmetry breaking. Due to the recycling effect (“continuous system renewals”31,32) provided by the crystal grinding mechanism, even small imperfections in that balance can often be tolerated (hence, the 1958

dx.doi.org/10.1021/cg200116e |Cryst. Growth Des. 2011, 11, 1957–1965

Crystal Growth & Design conversion is robust—by no means a “Goldilocks” scenario). However, akin to the case of high supersaturation in Kondepuditype conversions, very small particle size distributions (PSDs) produced by excessive attrition in Viedma-type conversions can similarly yield stochastic outcomes (negligible ee enhancement) for the deracemization process.25 Mechanisms Thought to Underpin Viedma-Type Processes. In this section, different (sometimes conflicting) views of the Viedma conversion are put forth in an attempt to highlight both the significant number of mechanisms involved in that process and the difficulty in identifying the essential one(s), without necessarily judging the relevance or importance of those ideas. The mechanism of “attrition-enhanced Ostwald ripening (OR)”4,5,25,29 continuously breaks apart the crystals, in an indiscriminate manner (i.e., both D and L mirror images, stochastically), thermodynamically favoring the evolution of a homochiral state provided that the solution racemization is not extremely rapid. It leads to the removal of “competing lineages...to leave just one common ancestor”,29 which, ultimately, results in the “achievement of homochirality in crystallizations, and, in general, the attainment of complete symmetry breaking in systems in which it operates”.29 In other words, under conditions of attrition-enhanced OR the enantiomer that has the majority fraction in the solid-state has the lower effective solution concentration. That scenario shifts the solution-phase equilibrium (racemization) to produce more molecules of the same handedness as the major crystal population (hence the “chiral amnesia”4,5,26), which can subsequently feed the growth of those crystals. Any depletion of molecules of the major solid-state enantiomer from the solution drives the dissolution and subsequent racemization of solids of the minor optical isomer via the coupled equilibria shown in Scheme 1. Viewed from a slightly different perspective, once a small enantiomeric imbalance has been achieved,33 the system begins to evolve toward solid-state homochirality (via nonlinear dynamics18,29) by a recycling effect whereby multiple mechanisms are sampled, continuously, until the most thermodynamically stable state—ultimately, a single homochiral crystal25—is achieved (note, however, that herein the conversion is considered complete when ∼100% ee is achieved, independent of the PSD). That recycling effect, whereby the system renewals are enhanced by the continuous crystal grinding, is consistent with the observation of dispersive kinetics8,31,32 that will be discussed later. The mechanisms involved in the process (refer to Scheme 1) include crystal nucleation, growth and/or OR, and denucleation (dissolution) of each enantiomeric crystal as well as the solution phase racemization of the two enantiomers. Since those processes are in near-equilibrium with each other, they all likely play a role in chiral symmetry breaking. As such, in the author0 s opinion, none of the mechanisms should be discounted on the grounds that they are not rate-limiting, without the kinetic data to support doing so. Crystal grinding by the glass beads allows the Viedma process to proceed faster than without attrition, since the breakage of crystals facilitates their dissolution (by increasing their surface area and hence decreasing their relative thermodynamic stability), consequently increasing the enantiomeric solution concentration (of increasingly the minor enantiomer fraction, as the conversion proceeds). While the rate enhancement possible from a solution phase concentration increase is clear from classical kinetics, with regard to the solid-state it is known from LifshitzSlyozovWagner (LSW) theory34,35 that smaller crystals (of the

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population that are sufficiently large to exceed the critical size needed for growth, as determined from the Kelvin equation) grow faster than larger ones, via OR, due to their higher surface energy. On the other hand, one must simultaneously consider the “surface area as a driving force for crystal growth”, since “the continued fragmentation of crystals by attrition also provides a relative increase in the surface area of the hand that has established an excess”.4 Presumably, such an enhanced surface area can also increase the probability of secondary nucleation of the major enantiomeric form in the solid-state. For the sake of completeness, it is pointed out here that Viedma-type conversions have also been described by an alternate mechanism3,23 in which subcritical fragments that are small enough to dissolve but large enough to retain their chirality are chipped from crystals by the stirring with glass beads. That chipping effect increases the dissolution of larger crystals, but it can be counter-acted by aggregation of the fragments with larger crystals of the same chirality. Alternatively, Saito and Hyuga36 recently proposed a model based on an enhanced enantiomorphic crystal surface reaction (however, that model might have been disproven, at least for certain cases, on the basis of experimental ee measurements27). Unfortunately, none of the (thermodynamic) arguments presented in this section allow one to pinpoint the (kinetic) rate-limiting step(s) in Viedma-type conversions. Opportunities for Kinetic Understanding. From Scheme 1 it is clear that Viedma-type processes involve several coupled equilibria. As such, the principle of microscopic reversibility37 might be38 important,39 which, in turn, makes it difficult to assign a priori a particular step as being rate-limiting. For example, Cartwright et al.29 showed that only by considering nucleation (primary and secondary) and OR, together, could ee values near unity be achieved in their simulations. Considering that variations in the system temperature and mixing/grinding conditions (Peclet number, amount/size/morphology/chemical nature of the grinding beads used, shear stress) can be used to tune the driving force for the evolution of homochirality, the potential exists for various mechanisms to emerge as rate-limiting as different system energetics are sampled. Once again, those mechanisms include the following (see Scheme 1): the solution phase racemization process, the dissolution of D crystals, the secondary nucleation of D crystals (remember that primary nucleation does not generate homochirality), the growth of D crystals, the OR of D crystals, the dissolution of L crystals, the secondary nucleation of L crystals, the growth of L crystals, and the OR of L crystals. Evidence of the presence of multiple mechanisms in Viedma-type conversions can be found in the recent literature. For example, by altering the experimental conditions, the kinetic profiles can be dramatically affected. Under different conditions of temperature and attrition efficiency (i.e., as different system energetics are probed), different mechanisms become rate-limiting. For example, “the form of the conversion profile obtained in the absence of enhanced attrition is altered significantly by temperature—at lower temperature, a steady, gradual conversion is observed rather than the typical autoinductive profile”.5 Note, however, that based on the system complexity, it is not recommended that nonisothermal kinetic profiles be compared to isothermal ones; similarly, different isothermal or mixing/attrition conditions can also potentially produce different rate-limiting mechanisms for the homochiral conversion. The OR mechanism is particularly intriguing for describing Viedma-type conversions from the viewpoint that, due to the 1959


Crystal Growth & Design GibbsThomson effect, crystals smaller than the critical size dissolve at the expense of the growth of larger ones. Such a mechanism supports amplification of the major solid-state enantiomeric form, and attrition clearly enhances that process, for reasons already mentioned. The Monte Carlo simulation work of Noorduin et al.,25 based on an attrition-enhanced OR mechanism, is quite convincing for several reasons. Chief among them, the empirical kinetic profiles (conversion transients, i.e., plots of ee vs time) that were generated appear to closely resemble the behavior observed experimentally.4,5 Second, the OR mechanism is well understood in terms of a classical, thermodynamic relation: the GibbsThomson/Kelvin equation. While the first point will be discussed further in the next section, with regard to the second observation, it is highlighted here that the Gibbs Thomson effect is just as important in OR as it is in Classical Nucleation theory (CNT).40,41 In CNT, the energy of the “critical nucleus” and, therefore, the activation energy barrier for nucleation is defined the same way; thus, nuclei that form that are smaller than the critical size are unstable and will denucleate (dissolve/dissociate), while those that are larger than the critical size will continue to grow. Clearly, the GibbsThomson effect does not provide definitive support of an underlying OR mechanism in Viedma-type conversions. It is important to note that OR cannot occur without the nucleation and growth of crystals and it, alone, cannot explain the evolution of homochirality.7 As the time-scales of OR and nucleation/crystal growth are typically quite different, those processes are often separable (OR typically takes place over much longer time-scales).42,43 Unfortunately, since the Viedma process has recently been demonstrated on the minute time-scale,7 in addition to the multiday experiments performed previously on the laboratory scale,4,5 the time-scale, alone, does not allow one a means by which to differentiate the two mechanisms. The use of deterministic models (avoiding simulations) to accurately/precisely describe the kinetics of Viedma-type conversions, thus realizing true mechanistic insight, has remained an elusive goal to this point. For instance, while both of the simulations in refs 25 and 29 yielded sigmoidal ee vs time (t) transients, which appear to adequately mimic experimental profiles,4,5 the classical first-order mechanism used to describe the kinetics over the entire conversion range25,26 often does an inadequate job because it relates a simple exponential (e.g., see Figure 10 in ref 25 and Figure 1 in ref 26). That observation suggests that a different model might be needed. Curiously, for the laboratory-scale systems studied in refs 4 and 5, the evolution of solid phase homochirality was shown to occur over the course of days/weeks, while in ref 26 the change in solution phase ee, as a function of t, was reported to occur much faster—over the course of only minutes/hours. That finding suggests that the solid-state kinetics might be rate-limiting.44 Furthermore, the faster kinetics observed on larger scale7 is also supportive of the idea of a ratelimiting physical process. Upon closer inspection, the sigmoidal kinetic transients corresponding to the solid fractions in refs 4 and 5 are visually similar to the “nucleation-and-growth” kinetics describable by the classical JohnsonMehlAvramiErofe0 evKolmogorov (JMAEK) equation4551 (eq 1, below) or, equivalently, by a fractal-time dispersive kinetics treatment.52 Of course, sigmoidal kinetics (“nonlinear dynamics”), which, in the “symmetrical S-shape” limit can be considered autocatalytic (for homogeneous processes), are thought to be important for asymmetric amplification in the homochiral evolution process,

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Figure 1. Simulated kinetic transients for a Viedma-type conversion, starting from different initial ee values (obtained from Figure 5 in ref 25). The data points were modeled using nonlinear regression fits to eq 6 (solid lines); see Table 1 for the relevant fit parameters and their associated errors (where applicable). For comparison, the broken line fit of the (2) data set was obtained using eq 1 (JMAEK model): R2 = 0.9870, k = (1.5 ( 0.9) 108 au, n = 4.64 ( 0.15.

as discussed earlier. The JMAEK (“Avrami”) equation can be written most simply as follows: p ¼ 1 ekt

n

ð1Þ

where, in a traditional solid-state phase transformation, p is the product crystal fraction in the system at a given conversion time, t; k is a pseudorate constant; and n is the system dimensionality (n = 2, 3, or 4 depending on whether the assumed simultaneous random/isotropic nucleation and fixed-rate growth—hence “nucleation-andgrowth”—of impinging spherical particles occurs in one, two, or three dimensions, respectively). For practical purposes, n is often determined empirically, since the stated dimensionalities usually do not provide optimal fits to empirical data. Fortunately, dispersive kinetics treatment52 can support that practice and any such values of n obtained via curve-fitting in the range 1 < n < 4 (whereby n = 1 represents the case of simple, first-order kinetics), although it does not provide much physical insight into most of those mechanisms.53 In any event, eq 1 generally relates so-called “stretched exponentials”. Dispersive Kinetics. Dispersive kinetics,8,31,32 sometimes referred to as “distributed kinetics”, is underpinned by the concept of a distribution of activation energies. The activation energy distribution (as opposed to the single activation energy that is typically inferred via inspection of the Arrhenius/Eyring equation) is simply a manifestation of system heterogeneity that can cause a reactant species to become either more or less reactive as a given conversion proceeds. Key examples are found in conversions that involve the solid-state, specifically, in nucleation and denucleation rate-limited processes (not all monomers in a given critical nucleus can be expected to have the same energy, nor do all the critical nuclei in a given phase transformation typically form at the same time). Such a change in the apparent reactivity often comes about as a result of local structural relaxation (nonequilibrium behavior or system dynamics) that occurs in parallel with the rate-limiting step of the conversion; the continuous environmental renewals can impact the kinetics if they occur on a time scale that is similar to, or slower than, the conversion under investigation. 1960


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In recent years, the author has developed his own approach for treating dispersive kinetics,8 relying on the assumption of an underlying MaxwellBoltzmann-like (M-B-like) distribution of activation energies (that fundamental assumption is investigated in a paper that is currently in preparation). The a priori definition of the general functional form of that underlying activation energy distribution was necessitated by his desire to circumvent the use of unitless (thus physically/mechanistically ambiguouse.g.53) parameters52 in those models. Note that, in terms of the known energy distributions in chemical physics, the M-B distribution inherently incorporates energy quantization into it, which the author thinks is important for describing processes (such as nucleation and denucleation) occurring on the nanometer scale because matter of that size is known to exhibit quantum confinement effects.54 Additionally, with regard to the other two frequently encountered energy distributions, namely the FermiDirac and BoseEinstein, both revert to the M-B distribution under the experimentally relevant conditions discussed in this work. Equation 2, below, is a dispersive kinetic model based on a firstorder conversion mechanism that has proven useful in describing “acceleratory”, sigmoidal pt transients common to nucleation ratelimited processes, including those of certain polymorphic interconversions.8 Acceleratory kinetics refer to asymmetrical, “S-shaped”, or “stretched exponential” transients in which the rate of conversion, postinduction period, occurs faster after the inflection point in the curve than before it (the opposite is true for deceleratory kinetics that typically involve processes that are denucleation rate-limited, such as some crystal dissolutions8). The derivation of the equation and its application to solid-state kinetics modeling was recently described elsewhere.8 R p ¼ 1 exp ð2Þ ½expðβt 2 Þ 1 t In eq 2, p and t are defined as before; however, for the purposes of this work, ee is used in place of p in both eqs 1 and 2 to model empirical data. That should not be seen by the reader as problematic, since p is directly proportional to the ee. Thus, to revert the ee values modeled in this work back into p values, one can simply use the following relationship: ee = 2p 1. For instance, when starting from a racemic mixture of conglomerate crystals where ee = 0, then p = 0.5 (unfortunately, since both eqs 1 and 2 operate on ordinate values ranging from zero, at t = 0, to unity, as t f ¥, they cannot be used directly to model data when (0, 0.5) is the starting point). The function in eq 2 can be described as an “exponential of an exponential”, as opposed to the simple exponential corresponding to a classical (nondispersive) first-order mechanism. The two fit parameters, R and β, have units of time and inverse-square time, respectively; they have been assigned the following physical interpretations, which reflect classical kinetic theories in the tindependent limit (where β = 0):8 R ¼ A1 e2 eEa

=RT

β¼

¼ A1 eΔH

ΔS‡ Rt 2

‡

=RT

ð3Þ ð4Þ

where A is an “Arrhenius-like” frequency factor with the entropic component removed from it (note that one can assume ΔS‡ = 2R at t = 0 since, from Transition State theory, Ea0 t Δ H‡ þ 2RT), Eais the “Arrhenius-like” (t-independent) portion of the

activation energy barrier, ΔH‡ is the corresponding activation enthalpy, and ΔS‡ is the t-dependent activation entropy (such that ΔG‡ = ΔH‡ TΔS‡, where ΔG‡ is also t-dependent due to the contribution from ΔS‡). In a crude sense, R is akin to a traditional rate constant that reflects the mean conversion rate, while β relates the distribution width (i.e., the dispersion about that mean specific rate). For acceleratory p vs t, or ee vs t, sigmoids, β is inherently positive (in eq 4), since it is responsible for producing the experimentally observed rate acceleration via a lowering of ΔG‡ as a function of t (the opposite is true for deceleratory conversions8). From eqs 3 and 4, it is possible to deduce the following parabolic t-dependence of the activation energy, Ea: Ea ¼ Ea RTβt 2

ð5Þ

Because of the simplicity of the author0 s model, eq 2, and the physical interpretation provided for each parameter, it is thought to be more insightful than the JMAEK model that has served as a mainstay in modeling nucleation-and-growth kinetics for the last several decades. In the next section, eq 2 will be compared to eq 1 in terms of its ability to precisely model the rate-limiting solidstate kinetic behavior of the evolution of homochirality in Viedma-type conversions. That is done in an attempt to elucidate the key mechanism in Viedma-type conversions, on the basis of a curve-fit comparison between the previously reported deterministic model (a first-order/simple exponential mechanism) and the “alternate kinetic models” discussed above, using characteristic transients taken from the recent literature. Note that, in all cases, the number of fit parameters in each model is limited to two. Modeling Solid-State Evolution of Homochirality Using Dispersive Kinetics. A recent paper25 used Monte Carlo simulation to show that the enantiomeric excess in Viedma-type conversions evolves as a function of time in a manner that yields asymmetric, sigmoidal ee vs t conversion transients. Those simulated transients, each starting from a different initial ee value, are reproduced here in Figure 1. The longest conversion (2) was found to be very well described by eq 2, assuming an initial racemate (in fact, the simulated data corresponds to an initial ee of 2.5%; see Table 1). From Figure 1 it can be seen that the effect of increasing the initial ee is mainly to translate the (2) curve from right to left along the abscissa; the ordinate intercept in each case appears to be very close to the initial ee value used in each stochastic simulation. Using the fit of the (2) curve by regression modeling with eq 2, it was possible to extract reasonable t estimates for the shifts corresponding to each (higher) starting ee value curve; those t-shifts are assigned the symbol c in the modified equation, below: R ð6Þ ½expðβðt þ cÞ2 Þ 1 ee ¼ 1 exp tþc The c values for each transient in Figure 1 are provided in Table 1. Using eq 6 and those c values, the remainder of the transients in the figure was fit. The good fits to each data set, over their entirety, support the use of eq 6 for this kinetic modeling application, over the classical first-order model (e.g. ref 25), which only treats a portion of each transient. In all cases, the values of R were found to range from 0.84 to 2.2 au (note that real time units cannot be assigned to the rate parameters due to the arbitrary assignment of the step durations in the original Monte 1961



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Table 1. Kinetic Parameters from the Regression Fits of the Data Sets in Figure 1 Using eq 6 and Predetermined Values of c (See Text for Details)a c R

β

set

initialbee

au)c

R2

(t, au)

(t2, au)

b

0.875

55.24

0.9971

0.84 ( 0.13

(1.56 ( 0.05) 103

O

0.750

52.10

0.9972

1.60 ( 0.16

(1.37 ( 0.03) 103

1 9

0.625 0.500

49.19 46.11

0.9994 0.9987

1.38 ( 0.06 1.11 ( 0.06

(1.427 ( 0.014) 103 (1.55 ( 0.02) 103

Δ

0.375

42.36

0.9987

1.94 ( 0.08

(1.288 ( 0.015) 103

0

0.250

37.23

0.9994

2.19 ( 0.06

(1.21 ( 0.01) 103

(

0.125

28.15

0.9978

1.57 ( 0.08

(1.205 ( 0.017) 103

)

0.050

16.00

0.9987

2.17 ( 0.07

(1.106 ( 0.011) 103

2

0.025

8.93

0.9987

1.33 ( 0.05

(1.119 ( 0.011) 103

data

a

(t,

b

au = arbitrary units. Values extracted from Figure 5 of ref 25. Parameters obtained using a regression fit of data set 2 to eq 2, assuming initial ee = 0; t = time. c

Carlo simulation) and the β values from 1.1 103 to 1.6 103 au. The variability in the two parameters is thought to originate mainly from stochastic differences across the transients, as noted in the original work.25 Of course, eq 6, like eq 1 and eq 2, inherently assumes that isothermal and constant-shear mixing conditions are maintained throughout. From Figure 1 (2), it is also evident that the two-fit-parameter-containing eq 1 (JMAEK model) does not describe the data nearly as well as the two-parameter eq 2. Since the author0 s model for denucleation (dissolution) rate-limited kinetics,8 which is complementary to eq 2, also did not model any of the transients presented in this work adequately (fits not shown), it is not discussed here further. From the earlier discussion and from the modeling work presented (thus far) in Figure 1, it is clear that the attritionenhanced OR mechanism presented by Noorduin et al.25 as the key factor in the stochastic evolution of homochirality in Viedmatype conversions might be challenged. From a purely kinetic viewpoint, the sigmoidal transients in Figure 1 are much better described by eq 6, a dispersive first-order model, than a traditional first-order one.55 That finding lends support to a nucleation rate-limited mechanism (e.g. ref 8). Interestingly, Uwaha and Katsuno28 recently reported the use of an extended Becker D€oring-type model to show that OR “is not the essential mechanism” and that “the direct crystallization of small chiral clusters and grinding are essential to the chirality conversion that leads to a homochiral state”. Their simulation work similarly produced kinetic transients with a generally sigmoidal profile, thought to reflect the “exponential amplification” in the ee evolution (e.g. ref 27). Fundamentally, one must be careful in modeling such transients because OR kinetics and nucleation rate-limited kinetics might be treatable in much the same manner. That is because both can be considered dispersive (stochastic) processes in which the critical cluster size, as related via the GibbsThomson/Kelvin equation, evolves over time as the system undergoes continuous renewals, causing the activation energy to change with t (e.g., see eq 5). Such similarity can lead to modeling ambiguity. As a case-in-point, a recent work56 used eq 1 to model the coarsening (OR) of gold nanoparticles; because of the strong empirical evidence for that kind of particle growth mechanism, yet, at the same time, noting the apparent

Figure 2. Experimental kinetic transients (data points obtained from ref 4) for the evolution of solid-state ee of an S-amino acid derivative in a Viedma-type conversion, starting from a mixture of equal proportions of crystals of each handedness, in the presence of different amounts of the additive, R-phenylglycinamide: 0.1% (Δ), 1.6% (1), 4.7% (O), and 8.7% (b). The solid lines are regression fits to the data points using eq 2: (Δ) R2 = 0.9999, R = (1.1 ( 0.4) 101 d, β = (4.0 ( 0.4) 101 d2; (1) R2 = 0.9996, R = 8 ( 4 d, β = (10 ( 4) 101 d2; (O) R2 = 0.9940, R = 2.2 ( 0.5 d, β = (1.37 ( 0.17) 102 d2; (b) R2 = 0.9938, R = 0.8 ( 0.2 d, β = (1.1 ( 0.1) 102 d2. For comparison, the (worse) broken line fits were obtained using eq 1 (JMAEK model); for two of the four data sets, a reasonable fit could not be obtained with that equation (curves not shown).

applicability of the JMAEK equation in fitting the observed sigmoidal kinetic transients, the authors of that paper deemed the behavior to be consistent with an “aggregative nucleation” mechanism. While the observation of nanoparticle aggregation is not uncommon in the literature (e.g. ref 57) nor is the concept of nucleation using subunits other than monomers a novel one (e.g. refs 5860), the best way to tell apart nucleation ratelimited kinetics from OR rate-limited kinetics might be to plot the evolution of the mean particle radius, Æræ, as a function of t. The cube of Æræ, corrected for the same power of the mean starting particle radius size (note that this is not the same as plotting the change in mean particle volume), evolves linearly as a function of t for diffusion-controlled OR cases; the same is true for the square of Ær æ in reaction-controlled OR cases,61 as predicted by LSW theory.34,35 With respect to the nucleation mechanism underlying eq 2, it will be shown in a future work that the t-dependence of Ær æ predicted by that dispersive kinetic model is very different from either OR case. Alternatively, one can examine the steady-state PSD profiles that are produced by OR versus nucleation rate-limited kinetics. In a paper currently under preparation by the author, it will be shown that while OR produces PSDs that are skewed from normal toward smaller particle sizes, for a nucleation rate-limited process the skew is in the opposite direction. That feature can also allow one to differentiate the two mechanisms.62 In order to determine whether eq 2 might be useful in realworld modeling applications (rather than just simulated transients) of symmetry-breaking, in Figure 2 some of the kinetic data pertaining to the attrition-enhanced solid-state ee enrichment of an amino acid derivative, obtained from ref 4, was fit using that dispersive kinetic model. From Figure 2 it can be seen that the evolution of the S enantiomer, driven by various relative amounts 1962



Figure 3. Isothermal (90 C) kinetic transient for the evolution of the solid-state ee of (a) L-aspartic acid and (b) D-aspartic acid in a Viedmatype conversion, starting from a nearly racemic mixture of crystals ( 0.993 in all cases. In Figure 3, data for the homochiral conversions of D- and Laspartic acid (separately), under conditions of constant grinding and isothermal heating (90 C), were modeled. Since the exact starting ee was not known (only that the conditions were “nearly racemic, 6 d [reflecting system stability, over shorter (nongeological) time periods] to those two plots in question, then the seemingly exponential transients can be visually transformed into asymmetric sigmoids, of the type shown in ref 4 and modeled here in Figure 2 (for isothermal conditions).

’ CONCLUSIONS Nucleation rate-limited processes that typically produce sigmoidal conversion transients8 have been modeled for decades with the JMAEK equation.4551 In this work, the sigmoidal transients obtained from both experiment and simulation4,5,25 of attrition-enhanced chiral symmetry breaking in the solid-state were modeled using the author0 s dispersive kinetic model for nucleation rate-limited conversions. That model, given by eq 2 (starting from racemate) or eq 6 (starting from some known initial ee), was found to be useful in describing the kinetics of Viedma-type conversions, a finding that is satisfying from three key standpoints. First, the model was found to fit the data better than the classical/nondispersive first-order mechanism used in the literature to-date25,26 (and better than both the JMAEK equation and, not shown, the author0 s dispersive kinetic model for denucleation rate-limited processes8). Second, the model fits yield meaningful kinetic quantities with physically relevant units. Finally, the nucleation rate-limited mechanism defined by eq 2 provides a kinetic amplification that is consistent with Frank0 s proposal18 yet simultaneously aligned with the basic thermodynamic arguments for amplification defined in the attritionenhanced OR mechanism25 (recall that the critical cluster size/ activation energy can be defined using the Kelvin equation in both CNT and LSW theory). The modeling successes presented herein associated with the use of eq 2 lend support to the notion that nucleation is the rate-limiting step in Viedma-type 1963


Crystal Growth & Design conversions. From previous literature, it is clear that the type of nucleation leading to the emergence of homochirality must be secondary (and not primary). As pointed out by a reviewer, more recent papers on breaking chiral symmetry often overlook (as this author did too, initially) much earlier work (e.g. ref 65) on the use of selective nucleation in systems involving more than one solute. That work demonstrated how control of nucleation could lead to separation of species, much in the same manner as this work describes separating enantiomeric forms, i.e., using secondary or “contact” nucleation. Selective nucleation (e.g. ref 66) and selective crystal growth (e.g. ref 67) approaches have continued to play an important role in process development to the present day. While it is not possible to create conditions conducive to OR without first nucleating (and growing) particles that exceed the critical nucleus size, which is typically on the order of nanometers in diameter (note that nucleation is often regarded as a lowprobability event at moderate supersaturations), there exists other important recent literature evidence that supports the idea that attrition-enhanced OR is not the major mechanism underpinning Viedma-type conversions.28 Thus, “attrition enhanced nucleation-and-growth” (as opposed to “attrition enhanced OR”), whereby secondary nucleation (with chiral recognition) plays the main role in the evolution of homochirality in the solid-state (as per Kondepudi-type processes1,20), is suggested here to be the key rate-limiting mechanism. OR is most likely to become important during the final stages of the conversion by facilitating the completion of the transformation, making it possible to achieve ee values approaching 100%, as shown by a previous work.29 In any event, the author believes future studies aimed at monitoring the PSD evolution (which is quite common in the current nanocrystal formation kinetics literature) during chiral symmetry breaking will prove valuable in verifying the ratelimiting (nucleation vs OR) mechanism. Energy input is necessary to provide a driving force for the chiral symmetry breaking as well as to establish a counterprocess24 (to the rate-limiting secondary nucleation of the major enantiomer in the solid-state) to produce system recycling that is necessary to achieve very high ee by minimizing the possibility of the system straying too “far from equilibrium”. Such thermodynamics allow a good energetic balance between the mechanistic pathways depicted in Scheme 1, which is thought to be important in achieving an optimal ee upgrade, analogous to the way in which a chromatographic column requires many partitioning cycles between the stationary and mobile phases in order to achieve baseline resolution for an enantiomeric pair in solution.10 What this work does not support is the following: (1) primary nucleation as a critical mechanism in Viedma-type processes (since it does not support the evolution of an ee enhancement starting from a racemic mixture of conglomerate crystals), (2) denucleation (dispersive dissolution) of the minor form as ratelimiting (as that would produce different-looking “deceleratory” kinetic transients, not “acceleratory” ones8,68,69), or (3) a classical (nondispersive) first-order mechanism25,26 for the emergence of ee, as a function of t, in the solid-state (due to the experimental observation of sigmoidal transients as opposed to simple exponentials). With regard to the recent proposal of a “Viedma ripening”27 mechanism, while it is true that attrition leads to the formation of a nonequilibrium PSD that might be different from the equilibrium case of traditional OR, this work and ref 28 des-

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cribe the formation of secondary nuclei/chiral clusters as the mechanistically critical step of the deracemization process. On the other hand, ref 27 appears to focus on the “reincorporation of small clusters” into crystals of the “major population in the solid phase”, essentially considering that process to be a new type of particle coarsening. Resultantly, some additional reconciliation might be needed here, perhaps via the proposal of an “aggregative secondary nucleation”-type mechanism. Of course, trying to account for nonequilibrium/relaxation effects in the transients was the main driver for this author0 s use of dispersive kinetic models in the present work. Furthermore, as mentioned above, distinguishing between a (dispersive) secondary nucleation mechanism and a (dispersive) ripening mechanism might benefit from additional experimental evidence. From an entirely thermodynamic perspective, the evolution of solid-phase homochirality in the Viedma process is controlled both by the solution phase racemization (following the Meyerhoffer double solubility rule,26,70 perhaps with catalysis by the crystal surface36) and by the OR mechanism. While one can take advantage of the former mechanism by introducing a small ee (i.e., a minor out-of-equilibrium perturbation) at the outset of the experiment to drive the evolution of homochirality toward the desired enantiomer (thus avoiding selection by random fluctuation, since the D and L homochiral states are energetically equivalent), the OR mechanism provides longer-term enhanced system stability (driving force) by decreasing the overall crystal surface area. The crystal grinding portion of the process provides a continuous energy supply for the disintegration of crystals which, in turn, facilitates their dissolution yet, simultaneously, increases the probability of both secondary nucleation or growth by OR (of mainly the major solid-state enantiomer, as the conversion proceeds) on their surfaces. Via crystal grinding effected by the use of glass beads, many cycles of crystal dissolution/nucleation/ growth/particle coarsening (i.e., “continuous system renewals”) are ensured. It is the resulting interplay/balance between the thermodynamics and kinetics (e.g. ref 2) that ultimately leads to “chiral amnesia”26 or, equivalently, “the common ancestor effect”29 that is critical to the high efficiency of these homochiral conversions.

’ AUTHOR INFORMATION Corresponding Author

E-mail: [email protected] or [email protected].

’ ACKNOWLEDGMENT The author is grateful for the specific recommendations of the (anonymous) journal reviewers who contributed to the final version of the paper. ’ REFERENCES (1) Viedma, C. Phys. Rev. Lett. 2005, 94, 065504. (2) Viedma, C. Astrobiology 2007, 7, 312. (3) Uwaha, M. J. Phys. Soc. Jpn. 2004, 73, 2601. (4) Noorduin, W. L.; Izumi, T.; Millemaggi, A.; Leeman, M.; Meekes, H.; Van Enckevort, W. J. P.; Kellogg, R. M.; Kaptein, B.; Vlieg, E.; Blackmond, D. G. J. Am. Chem. Soc. 2008, 130, 1158. (5) Viedma, C.; Ortiz, J. E.; de Torres, T.; Izumi, T.; Blackmond, D. G. J. Am. Chem. Soc. 2008, 130, 15274. (6) Soai, K.; Shibata, T.; Morioka, H.; Choji, K. Nature 1995, 378, 767. 1964


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Kinetics and Thermodynamics of Efficient Chiral Symmetry Breaking

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