Experimental and Theoretical Study of the Emergence of Single

Synopsis. Experimental and theoretical studies help to deconvolute the driving forces for crystal growth during attrition-enhanced deracemization, dem...
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Experimental and Theoretical Study of the Emergence of Single Chirality in Attrition-Enhanced Deracemization Dragos Gherase,† Devin Conroy,‡ Omar K. Matar,‡ and Donna G. Blackmond*,†,‡ †

Department of Chemistry, The Scripps Research Institute, La Jolla, California 92037, United States Department of Chemical Engineering, Imperial College, London SW72AZ, U.K.

‡

S Supporting Information *

ABSTRACT: Experimental studies help to deconvolute the driving forces for crystal growth during attrition-enhanced deracemization, demonstrating an interplay between crystal size and crystal number in the emergence of homochirality. A semiempirical population balance model is presented based on considerations of the solubility driving force, as outlined by the Gibbs−Thomson rule, and a frequency factor based on the total interfacial surface area between solid crystals and the solution phase.

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INTRODUCTION The single-handedness of biological molecules has fascinated scientists since Pasteur’s first painstaking physical separation of the mirror-image crystals of a tartrate salt.1 The emergence of single chirality requires sustaining and propagating a small imbalance induced by the initial symmetry breaking in a presumably racemic prebiotic environment. Theoretical models for how single chirality might have evolved have been discussed for more than half a century,2 but only more recently have experimental studies begun to address the question of chiral amplification more directly. The Soai autocatalytic reaction3 and Kondepudi’s “Eve crystal” model4 offered the first experimental proof-of-concept for the evolution of homochirality from a stochastically induced imbalance in a far-fromequilibrium scenario. A mathematical description of the Soai reaction was subsequently developed from kinetic, spectroscopic, and modeling investigations by Blackmond and Brown.5 At the other end of the spectrum, a purely equilibrium scenario for solution-phase enantioenrichment has been proposed in experimental and theoretical investigations of the phase behavior of amino acids.6−8 A significant departure from both equilibrium and far-fromequilibrium models for enantioenrichment arose with Viedma’s9 striking demonstration that a racemic mixture of enantiomorphic crystals of the achiral salt NaClO3 under dynamic phase equilibrium evolves inexorably to crystals of a © 2014 American Chemical Society

single solid enantiomorph when mechanically stirred in the presence of glass beads (Scheme 1). This landmark finding was Scheme 1. Emergence of Solid-Phase Homochirality

extended by Blackmond and colleagues to intrinsically chiral molecules including amino acids10 and their derivatives.11 In this case, the molecular interconversion of enantiomers in the solution phase serves as the conduit for the net movement from one enantiomorphic solid to the other. The term “chiral amnesia”12 was coined to highlight the key, if paradoxical, role Received: July 15, 2013 Revised: January 23, 2014 Published: February 3, 2014 928

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Models for Homochirality. The sigmoidal profile of the temporal change in CEE observed in attrition-enhanced deracemization processes, as shown in Figure 1a,18 has been

played by solution phase racemization; it is the loss of a molecule’s solid-phase chiral history that drives the evolution to homochirality. The term “Viedma ripening”13 has also been employed to describe the phenomenon. To rationalize the emergence of a single chiral solid from racemic mixtures, Viedma14 and Blackmond12 have each proposed that the process is kinetically driven by crystal sizedependent solubility attributed to the Gibbs−Thomson rule (eq 1). In any collection of crystals of varying sizes in dynamic exchange with its solution phase, growth or dissolution of a crystal (eq 2) occurs according to the difference between the instantaneous solution concentration, C0 and that crystal’s solubility, Csol R (for a crystal of radius R). Any particle for which C0 > Csol exhibits a driving force for crystal growth by addition R of molecules to the crystal from solution; for particles where the converse is true, C0 < Csol R , molecules are lost from the crystal to the solution. CRsol =

⎛ 2γVm 1 ⎞ Cm ·exp⎜ · ⎟ 2 ⎝ KBT R ⎠

GR = k·(C0 − CRsol)

(1) (2)

where Csol R = solubility of a crystal of radius R, Cm = infinite planar solubility, C0 = solution concentration, γ = surface tension, Vm = molecular volume, KBT = Boltzmann constant × absolute temperature, GR = net crystal growth rate, and k = rate constant for interfacial mass transfer. We recently showed that a crystal size-induced imbalance in the solubilities of two enantiomorphs of different average sizes may be employed to effect attrition-enhanced deracemization under conditions where the two enantiomorphic solids are in separate flasks connected by their common solution phase. Under solution racemization conditions, molecules dissolving from smaller crystals of one enantiomorph convert to the other enantiomer in solution and then may reaccrete onto larger crystals of the other enantiomorph. The net movement from one (smaller crystal size) enantiomorphic solid to the other (larger crystal size) continues as dictated by the Gibbs− Thomson rule until one solid is entirely depleted. This solubility imbalance induced by crystal size differences is analogous to Dimroth’s principle15 for isomeric compounds that exhibit intrinsically different solubilities, which provides the fundamental basis of preferential crystallization, a process that is routinely applied to achieve selective production of diastereomeric compounds in the pharmaceutical industry.16 A point of considerable discussion has been whether the solubility considerations of the Gibbs−Thomson rule as outlined above provide a sufficient rationale for the experimental observations of the emergence of a single solid enantiomorph, including both the direction of the chiral outcome and the sigmoidal profiles of crystal enantiomeric excess (CEE) exhibited over time.17 In this report, we present detailed experiments and we describe a modified population balance model aimed at quantifying and deconvoluting the driving forces that contribute to attrition-enhanced deracemization. These findings demonstrate that the sense of the evolution to solid phase homochirality may be accurately predicted by understanding the kinetic and thermodynamic processes associated with crystal size-dependent solubility without invoking additional processes such as agglomeration of crystals or subcritical clusters.

Figure 1. Experimental and simulated profiles for the evolution of homochirality. (a) Experimental data for crystal ee (CEE) from attrition-enhanced deracemization illustrating sigmoidal behavior.18 (b) Simulation of Wright−Fisher model (eq 3) for genetic drift in a population of two alleles, which does not exhibit sigmoidal behavior.20 (c) simulation of Decker model (eq 4) for autocatalytic reaction with asymmetric amplification exhibiting sigmoidal behavior.21

the subject of much discussion. It has been noted that models based on stochastic population genetics do not exhibit such behavior.19 The simplest approach traditionally used to rationalize “genetic drift” in a population of equal amounts of two alleles, or alternate forms of a gene, is the Wright−Fisher model.20 The evolution of such a system is described in this model in terms of the number of generations required for “segregation,” which occurs when the system arrives at a homogeneous population containing only one or the other allele (eq 3). Comparison of the temporal form of an attrition enhanced deracemization experiment (Figure 1a) to that of the Wright−Fischer profile (Figure 1b) reveals that this model does not capture the experimentally observed behavior. Such a simple description of stochastic genetic drift is clearly not 929

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sufficient to explain the evolution of crystal enantiomeric excess. T̅(seg) = N · (− 4· (x ln(x) + (1 − x)ln(1 − x)))

While eq 5 captures the sigmoidal form observed in attritionenhanced deracemization, it is clear that the analogy between chemical reaction rate laws and cluster-crystal interactions is an imperfect one. Chemical reaction kinetics is grounded in fundamental principles of collision theory, while such rate laws applied to the deracemization process are purely empirical; chemical reaction rate constants have fundamental meaning, while the rate constants obtained from a fit of such a model to deracemization data must be considered as empirical adjustable parameters. Thus a fit of experimental data to the model cannot be claimed as confirmation of the role of clusters in attritionenhanced deracemization. The existence of subcritical clusters in supersaturated solutions undergoing primary nucleation has been documented for both organic and inorganic materials.23,24 Recently, elegant experiments have documented enantiomer-specific attachment between macroscopic crystals in warmed, shaken solutions.25 However, the viability of such processes at the near-equilibrium solution concentrations and under the demanding physical attrition conditions of Viedma ripening remains an open question, and no experimental confirmation has yet been demonstrated. Our recent demonstration of the emergence of single solid phase chirality utilized an experimental design that isolated the two enantiomorphs in separate pots with different stirring rates connected by filters to prevent solid-phase mass transfer. Although it could be argued that subcritical clusters of molecules might pass through the filters, this design clearly highlights the critical role of differential solubility in the deracemization process. A separate modeling approach based on population balance modeling was reported by Iggland and Mazzotti.26 An important advantage over other models is incorporation of the physical principles of the Gibbs−Thomson rule along with mathematical descriptions of crystal attrition and agglomeration. Agglomeration is a second-order process leading to chiral amplification similar to the inclusion of cluster agglomeration with crystals in the Uwaha chemical reaction model. However, in the population balance model crystal agglomeration occurs not only for subcritical clusters of molecules but between bona fide crystals. Under the attrition conditions of the deracemization process, it is not clear that either crystal or subcritical crystal coalescence processes would be operable. Considerations of Crystal Size versus Crystal Number. Another aspect of attrition-enhanced deracemization that has not been adequately rationalized by the models proposed to date is the role that crystal size vs crystal number plays in the direction of evolution of homochirality. While most experimental observations have found that, for two populations of enantiomorphic crystals of approximately the same crystal size distribution, the crystal enantiomeric excess evolves toward the enantiomorph that is in excess at the outset, the opposite has also been shown (Figure 2). Figure 2a (magenta circles, replotted from ref 27) shows an example of the conventional behavior is shown for compound 1a, where homochirality evolves to the major enantiomorph when the two exhibited identical crystal size distributions at the outset. Kaptein et al27 showed the first example of the contrary case, where homochirality for 1a evolved toward the enantiomer that was present at the outset as a minor population of larger crystals (Figure 2a, blue circles). The example in Figure 2b from our work using compound 1b28 corroborated these findings of a subtle role for both size and number of crystals. Quantitative measurements of initial crystal sizes and

(3)

where T̅ (seg) = time (mean generations) to homochirality in units of sample size, x = fraction of one allele (enantiomer), and N = population size. The most commonly discussed chemically based rationalizations of the evolution of homochirality focus on asymmetric amplification in autocatalytic reactions, processes that do exhibit sigmoidal profiles for enantiomeric excess as a function of time (Figure 1c). A key feature of autocatalytic models that leads to asymmetric amplification is an enantiomer concentration ratio that does not scale linearly with the ratio of their replication rates. For example, in the Decker model shown in Scheme 2,21 the ratio of rates of self-production of L and D Scheme 2. Decker Autocatalytic Reaction Model for Chiral Amplification

enantiomers from substrate A is proportional to the square of their concentration ratio (eq 4). This higher-order process is required in order to effect amplification of enantiomeric excess in autocatalytic reactions, and analogies have been made to chiral amplification in attrition-enhanced racemization. rate(L) [L]2 = rate(D) [D]2

(4)

Because autocatalytic reaction models such as that shown in Scheme 2 and Figure 1c give chiral amplification profiles that resemble those for attrition-enhanced deracemization, such chemical models have been extended to describe this physical process by treating the rate of crystal growth and dissolution as mathematically analogous to elementary chemical reaction steps, as shown in Scheme 3. Uwaha22 proposed for the rate of Scheme 3. Simplified Uwaha Model for Attrition-Enhanced Deracemization Based on Chemical Reaction Models

addition of a cluster of molecules of one enantiomer (Dcluster or Lcluster) to a crystal of the same hand (Dcrystal or Lcrystal) may be considered as mathematically equivalent to a bimolecular reaction between the two. Treating the addition of a cluster of molecules in one step, rather than consecutive addition of single molecules, provides a higher-order process with concomitant sigmoidal amplification analogous to that observed from eq 4. Coupled with the solution interconversion of enantiomers, this provides a mathematical description of the evolution of one crystal enantiomorph at the expense of the other. rate(Dcrystal ) rate(Lcrystal )

∝

[Dcluster ] [Dcrystal ] · [Lcluster ] [Dcrystal ]

(5) 930

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circumstances where the larger crystals are far fewer in number than the smaller crystals. The experimental results thus demonstrate that both factors, crystal size and number of crystals of each enantiomorph, can influence the ultimate stereochemical outcome. The interplay between these two parameters may be likened to the interplay between kinetics and thermodynamics, as outlined in Scheme 4. Scheme 4. Processes Affecting Net Rate of Crystal Growth for (a) Large and (b) Small Crystals

In addition to the concentration driving force given by the Gibbs−Thomson rule, the net rate of crystal growth also depends on the frequency of contact between solution molecules and solid crystals, which is proportional to total exposed crystal surface area and increases with decreasing crystal size. The concentration driving force dictated by the Gibbs−Thomson relationship may be thought of as a thermodynamic parameter, while the role of total exposed surface area may be thought of as a frequency factor, or a kinetic parameter, incorporating surface area into the mass transfer coefficient in expressions for the rate of crystal growth/ dissolution.29 As shown in Scheme 4, these two processes influencing crystal growth correlate in opposite ways with crystal size. Thus a larger number of smaller crystals may exhibit a faster growth rate in a concentrated solution than do a smaller number of larger crystals, even though the latter crystals have an intrinsically greater thermodynamic driving force for crystal growth. In any particular case, the question of which driving force dominates, that is, “size versus number” and must be addressed. The aim of the present work is to quantify further the influence of the two parameters of crystal size and crystal number on the attrition-enhanced deracemization. We carried out detailed experimental measurements in order to deconvolute the two terms shown in Scheme 4 in the equation for the net rate of crystal growth in this process. These measurements are combined with a modified form of the population balance model developed by Mazzotti26 that allows a crystal size dependence of the mass transfer coefficient, presenting a physically reasonable approach to accounting for the question of “size versus number”. We use these results to address the question of whether agglomeration of crystals or subcritical

Figure 2. Evolution of crystal enantiomorphic excess (%CEE) during attrition-enhanced deracemization showing the varied relationships between initial CEE, initial crystal size, and the sense of chirality of the emergent enantiomorph. (a) Compound 1a, plot redrawn from the data in ref 27, and (b) compound 1b, plot redrawn from the data in ref 28.

solubilities showed that the homochiral outcome may be decided either by crystal size (larger size of one enantiomorph overcomes fewer total molecules) or by crystal number (larger number of molecules of one enantiomorph overcomes smaller average crystal size). A threshold CEE value exists at which the deciding factor switches between the initial bias in CEE and the initial bias in crystal size. Simple kinetic reaction-inspired models such as the Uwaha model of eq 3 cannot account for the results of Figure 2, and indeed such models do not consider effects of crystal sizeinduced solubility differences. By contrast, the population balance model developed by Mazzotti26 documented one set of conditions where the initially minor population of crystals evolves to homochirality, even though the model clearly dictates that, at any snapshot in time, the larger crystals will grow at the expense of smaller crystals. Agglomeration of smaller crystals of one hand effectively switched that enantiomorph to become the larger crystals during the deracemization process, thus allowing its evolution to homochirality. Such an outcome is arguably not physically reasonable under all circumstances, for example under 931

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clusters with crystals is a requirement of the attrition-enhanced deracemization process.

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RESULTS AND DISCUSSION Experimental Measurements. The results of previous studies presented in Figure 1 indicate that the interplay between the parameters of crystal size and crystal number must be considered in order to understand the deracemization process. Measurements to quantify the role of these parameters were carried out for the conglomerate system of compound 1b. We have previously shown different size distributions of 1b may be prepared by stirring slurries of crystals in the presence or absence of glass beads at different stirring or sonication rates. We further demonstrated that average size distribution is correlated to solubility. Thus for the current studies, we prepared populations of particles with different average crystal sizes and solubilities by carefully controlling the stirring/ attrition/sonication conditions in the absence of solution phase racemization. Two examples of pure solid enantiomorphs, designated “large” and “small” crystals, exhibiting solubilities in mg/mL of 7.3 ± 1.3% and 7.7 ± 1.8%, respectively, were prepared, for a ∼6% difference in solubility between the large and small crystals. These solubilities were measured for each enantiomorph as large and as small crystals. A third set of conditions employing even more vigorous stirring/attrition conditions with rac-1b to yield a racemic solution with a concentration exceeding the solubility of either the large or the small crystals described above. In this case the slurry was sonicated until the entire racemic solid dissolved to yield a highly concentrated solution of rac-1b at 18 mg/mL or 9 mg/mL for each enantiopure solid according to the Meyerhoffer double solubility rule. Since this concentration is higher than the solubility of either the “large crystal (S)-1b” or the “small crystal (R)-1b” prepared as described above, mixing either of these enantiopure solids in the concentrated racemic solution should result in net crystal growth, or net movement of molecules from the solution to the solid in both cases. The use of a concentrated racemic solution affords an elegant way to use enantiomeric excess measurements and compare the uptake of molecules from solution onto the large and small crystals. The rate and number of molecules deposited onto each enantiopure solid from the initially racemic solution may be calculated by measuring the resulting deviation of the solution phase enantiomeric excess from racemic after a given time, since only molecules of the same hand will deposit on that solid. Uptake of molecules of S-(1b) onto S-crystals from the racemic solution will result in a solution ee toward the Renantiomer, while uptake of molecules of R-(1b)onto R-crystals will give a solution ee toward S. Solid samples of large crystal (S)-1b and small crystal (R)-1b of equal weight were added to two separate vials containing equal volumes of the concentrated solution of rac-1b as shown in Figure 3. Each vial was shaken mildly for 5 min after which time the solid was removed by filtering. The enantiomeric excess of the remaining solution was then measured. The results of these two experiments are depicted in Figure 3a and b.30 The larger deviation from racemic of the final solution ee in Figure 3a compared to Figure 3b shows that the large crystals took up more molecules than the small crystals. This confirms that the larger crystals have a greater driving force for adding molecules, as predicted by the Gibbs−Thomson rule. Complementing these two experiments are those shown in Figures 4a and 4b where large and small enantiopure crystals

Figure 3. Phase transfer of molecules from a concentrated solution of rac-1b (18 mg/mL) to enantiopure crystals as measured from the deviation in solution ee from racemic. a) large crystals of (S)-1b, solubility 7.3 ± 1.3% mg/mL and b) small crystals of (R)-1b, solubility and 7.7 ± 1.8% mg/mL. Equal weights of crystals were used in the two experiments. Solution ee values measured by chiral HPLC after 5 min mixing.

Figure 4. Competition between large and small crystals for molecules from a concentrated racemic solution phase: (a) 1:1 mixture by weight of small (R)-1b and large (S)-1b and (b) 2:1 mixture by weight of small (R)-1b and large (S)-1b. Solution ee values measured by chiral HPLC after 5 min mixing. Small and large crystal solubilities and racemic solution concentrations as in Figure 3 932

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compete in the same vial for molecules in a concentrated solution. Figure 4a shows that a slight excess toward (R)-1b results when equal weights of the two enantiomorphs are mixed in the racemic concentrated solution. Under these conditions the concentration driving force of the large crystals removing (S)-1b from solution outweighs the larger surface area provided by the smaller crystals of (R)-1b. The racemic final solution ee of Figure 4b demonstrates that doubling the amount of the smaller crystals allows the larger solubility driving force of the larger crystals to be balanced by the greater available surface area of the smaller crystals. These measurements capture the instantaneous initial rate of movement of molecules from an enantiomer in a solution at higher concentration onto a solid of the same hand. A critical point is that these mass transfer processes are documented in the absence of the conditions required for attrition-enhanced deracemization: no solution-phase interconversion between enantiomers occurs, and neither attrition nor agglomeration of the crystals is expected during the mild stirring that delivers molecules from the slightly supersaturated solution to the crystals of different sizes. Thus the subcritical clusters of molecules that are postulated to be required for rationalization of attrition-enhanced deracemization are not expected to exist under the conditions of these experiments. An important remaining question from the studies carried out in this work, therefore, is whether the quantitative mass transfer processes from solution phase molecules to solid phase crystals, as outlined in the experimental measurements of Figures 3 and 4, in the absence of influences from attrition, agglomeration, or subcritical cluster formation, can fully predict the outcome of attrition enhanced deracemization experiments carried out by combining different amounts of the small and large crystals used in the experiments of Figures 3 and 4. To address this question, experiments were carried out under conditions where the physical mass transfer rate processes between solid enantiomorphs and solution molecules outlined in Figures 3 and 4 are accompanied by the chemical rate process of solution racemization and the enhanced attrition caused by the physical grinding of the solids. During attrition-enhanced deracemization, the interconversion between enantiomeric molecules in solution allows a net rate of movement of molecules from one enantiomorph to the other to be sustained until solid phase homochirality is achieved. The results of Figure 4 predict that movement from the smaller R crystals to the larger S crystals is expected for a deracemization experiment under the conditions shown in Figure 3a, with equal weights of each enantiomorph, but that homochirality toward the smaller R crystals is predicted when a 2-fold or greater excess of the smaller crystals is employed, as in Figure 4b. This prediction is outlined in Figure 5 and is tested in the experiments whose results are reported in Figure 6. Figure 6 shows the results of three attrition-enhanced deracemization experiments in which three separate vials are prepared in which a slurry of solid rac-1b in a saturated solution containing catalytic DBU is stirred with glass beads. Under these conditions, the two enantiomorphs exhibit identical crystal sizes. Each vial is then seeded with different proportions of large (S)-1b crystals and small (R)-1b crystals for a total of ∼5% total solid seed added in each experiment. Figure 6 shows the evolution of solid phase enantiomeric excess as a function of time for cases where the small (R)-1b/large (S)-1b seed weight ratio is 1:1, 2:1, and 5:1. For equal seed weights, the chiral outcome is dictated by the large crystals, while the small

Figure 5. Predicted outcomes for attrition-enhanced deracemization initiated with varying relative amounts of large and small seed crystals of each hand.

Figure 6. Experimental outcome of attrition-enhanced deracemization under three different conditions of solid crystals seeding. Solid phase ee (%CEE) is shown as a function of time during attrition-enhanced deracemization initiated with varying relative amounts of small R and large S crystals of 1b (see Scheme 4). Small and large crystals are defined by their solubilities given in Figure 3. The initial %CEE for the solid phase in these three experiments, including the racemic solid and the added seed, are 1:1 = 0% CEE; 2:1 = 1.6% CEE (R); 5:1 = 3.19% CEE (R).

crystals dictate the sense of homochirality when small crystals are in excess of 2:1 and 5:1. This result is in excellent agreement with the prediction shown in Figure 5 made from the solubility studies of Figures 3 and 4. The results in Figure 6 show that the outcome of attritionenhanced deracemization experiments is successfully predicted by simple consideration of the relative solubilities of different crystal sizes. The results are fully rationalized by consideration of the combined effects of the kinetic and thermodynamic driving forces described in Scheme 4. Quantitative assessment of the interplay between crystal size and crystal number allows prediction of the outcome of any attrition-enhanced deracemization experiment. Additional driving forces, such as that provided by the agglomeration of subcritical clusters of molecules, are not required to rationalize the emergence of solid-phase homochirality. Mathematical Model. Population balance analysis a growing field of mathematical modeling that is concerned with the ways in which particles in a given population can grow, disappear, and interact with one another such that the particle population redistributes itself over time. Such models have 933

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distribution in terms of solid volume V (eqs 7 and 8). Thus the net growth rate of eq 6 takes into account both of the driving forces shown in Scheme 4, the concentration driving force and the frequency factor related to total exposed surface area.

been applied to aspects of crystallization processes including the analysis and prediction of the performance of impact grinding mills.29,31 Attrition-enhanced deracemization was first subjected to analysis using a population balance model by Iggland and Mazzotti.26 This model effectively captures many of the experimental observations, and it offers deeper insight into the molecular mechanism of the process compared to other simpler models because it incorporates the Gibbs− Thomson driving force of crystal size-induced solubility differences. However, the size of the smallest “clusters” of molecules that are not bona fide crystals but contribute to the agglomeration process was not well-defined. We developed a population balance model slightly modified from that treatment to refine the physical description of the interplay of crystal size and crystal number in attritionenhanced racemization as illustrated in Scheme 4, specifically by introducing the “frequency factor” as a surface area dependence of the mass transfer coefficient. In the Mazzotti model, the mass transfer coefficient analogous to k in eq 2 in this work is treated as a constant for all crystal sizes. The surface area dependence of mass transfer coefficients has been employed in models of gas−liquid mass transfer32 as well as liquid solid mass transfer, and in particular for dissolution kinetics in a population balance framework.29 The Mazzotti model addresses the issue of size versus number by allowing the enantiomorph with smaller crystals at the outset to be converted to the enantiomorph with larger crystals via agglomeration of crystals, with different agglomeration rates being assigned to interactions between large−large, small−small, and large−small crystals. Our aim in modifying the population balance model was to assess whether the trends observed experimentally for the emergence of solid-phase homochirality influenced by crystal size and crystal number could be described by solubility considerations in such a model without invoking the coalescence of crystals or subcritical clusters of molecules under attrition conditions, by inclusion of terms to account for the kinetic or frequency term related to crystal surface area as illustrated in Scheme 4. Since the experimental studies presented here accurately predict deracemization outcomes from kinetic solubility measurements in the absence of conditions conducive to agglomeration phenomena, we wanted to probe whether a semiempirical population balance model based on the same tenets might also provide an accurate description of the observed behavior. The model employs numerical integration of population balance equations that describe the net growth of crystals in a well-stirred vessel where dissolution of molecules from, and reaccretion of molecules onto, enantiomorphic crystals in contact with their solution phase occurs concomitant with enantiomer interconversion in solution via racemization and where crystals break because of grinding with glass beads. The solid phases are simulated as a Gaussian size distribution of constant density spherical crystals, each containing only molecules of one or the other enantiomer D or enantiomer 33 L. Eq 6 describes the growth rate Gi(R) rate for crystals of a given radius R, where i = D or L. The term containing the solubility difference reflects the Gibb−Thomson driving force of eq 1. The term containing Γ incorporates the surface area dependence into the mass transfer coefficient, as a frequency factor related to the probability of encountering a specific crystal of size R. This term is proportional both to the available crystal surface area for all particles of size R and the particle number concentration, f i, based on a temporal population

Gi , R = k′·(1 + Γi , R) ·(Ci ,0 − CRsol)

i=

D

or L

Γi , R = k″·fi ·R2

fi (R , t ) =

(6) (7)

ni(V , t )dV dR

(8)

The equation for Γ accounts for the probability that a solution phase molecule may encounter a range of sizes of likehanded particles all meeting the criterion Ci > Csol,R that creates a positive driving force toward reaccretion of the molecule onto a crystal. Thus the instantaneous growth rate of crystals of size R may exceed that for crystals of a larger size if the total available surface area of the smaller crystals counterbalances their smaller solubility driving force. Equation 6 for net crystal growth rate may be considered as an analogy to the Curtin− Hammett principle, which demonstrates that the overall rate of a chemical reaction depends on both the reactivity and the stability of a reacting species. In the present case, both kinetics (probability of encounter) and thermodynamics (solubility driving force) are allowed to contribute to the evolution of crystal size in these dissolution/reaccretion processes. The role of attrition by glass beads is expressed in the temporal change in the particle number concentration (eq 6). This expression contains a term Si(R,R′) describing the frequency of crystal breakage and the mean number of daughter particles formed in a breakage event, expressed as a function of the kinetic energy and time scale of collisions between D or L particles and glass beads. Although particle agglomeration is excluded in some of our simulations, a term describing this but may be included, expressed by Qi(R,R′) that allow for crystal growth because of coalescence of crystals. dfi dt

+

∂ (f ·Gi) = Si(R , R′) + Q i(R , R′) ∂R i

(9)

A minimum crystal size is set under which a crystal dissolves completely, releasing all molecules to the solution phase and altering the solution concentration accordingly. No new crystals are permitted to form directly from primary nucleation of molecules in solution, in accordance with experimental observations. The chemical interconversion between D and L enantiomers in the solution phase is described by the first order racemization reactions of eq 10. The total mass of D and L molecules summed over the solution phase (volume = Vliq) and solid phases (solid phase density = ρs) is given by eq 11 for each enantiomer. While D and L molecules may interconvert, and each enantiomer’s relative fraction of the total mass may change over time, the total mass in the system is conserved (eq 12). k rac

D XooY L k rac

dC D = −k rac(C D − C L) dt

mi(t ) =

∫ Ci dVliq + 43π ·ρs ·∫0

mL + mD = mtotal = constant

∞

(10)

(R3·fi )dR

(11) (12)

These equations are rendered dimensionless for carrying out numerical integration of the differential equation describing net 934

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crystal growth rate for each crystal size in the population as a function of time. The development and nondimensionalization of these equations is described in detail in the Supporting Information. The results of a simulation of the evolution of solid phase CEE as a function of time applying this mathematical model is shown in Figure 7 for a case where the initial crystal size

Figure 8. Population balance model showing the evolution of solid phase enantiomeric excess as a function of time for different initial crystal enantiomeric excess (CEE) for a system with L crystals exhibiting a larger initial average crystal size (ΔRav = 6.4). Initial solidphase ee values, proceeding counterclockwise from lower left: CEE(0) = −5%, 0%, 4.9%, 5.0%, 5.1%, 5.2%, 7.5%, and 10%. Profiles in red: Initially major enantiomer is converted to the initially minor enantiomer over time.

Figure 7. Evolution of the particle number concentration f i and CSDs for CEE0 = 5% (D) and the initial size distribution shifted slightly toward D crystals. Top: D. Bottom: L. Figure 9. Population balance model showing the evolution of solid phase enantiomeric excess as a function of time for mixtures of crystals with initial CEE(0) = 5% CEE, where the minor enantiomer exhibits initial Gaussian distributions of crystals with average sizes as shown in Figure 1. The difference (major − minor) in average initial crystal size (arbitrary units), proceeding clockwise from top left: 6.4, 0, −3.8, −5.1, −6.4, and −12.7. Profiles in red: Initially minor enantiomer exhibits larger average initial crystal size.

distribution is shifted slightly toward the major enantiomorph at CEE0 = 5% (D). The agglomeration term in eq 9 was set equal to zero and the racemization rate was taken to be very fast such the CD ≈ CL at any time, and the particle breaking function in eq 9 was fixed to give two daughter crystals of equal mass. Figure 8 shows curves of CEE versus time for a series of simulations of attrition-enhanced deracemization carried out using a range of different initial CEE values and a fixed initial difference in the average radius of the L crystals compared to the D crystals. Figure 9 shows CEE versus time for a series of simulations employing a fixed initial ee value with different initial crystal size distributions of L and D crystals. Our aim was to probe the form of the temporal ee profile for simulations in the absence of the agglomeration of crystals as well as to explore realistic conditions in which the greater number of smaller size crystals of one enantiomer drives the evolution of homochirality, as well as the inverse case, where a smaller number of molecules exhibiting larger crystal size dominates the process. Both Figures 8 and 9 reveal that the model describes the evolution of homochirality as a sigmoidal function of time in all

cases, without inclusion of a contributing role for coalescence of crystals in the process. Thus solubility concepts of the Gibbs− Thomson rule for either a small imbalance in crystal size distribution or number molecules of each handedness provide the driving force that describes an accelerating process toward homochirality. Figure 8 shows that at lower initial % CEE values, the hand with the larger excess is converted to the minor hand exhibiting the larger particle size. At higher % CEE values, the hand with smaller crystals but larger enantiomeric excess decides the outcome for solid phase homochirality, Figure 9 shows that the sense of solid phase homochirality becomes controlled by the minor enantiomer fraction exhibiting the larger average initial 935

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recent population balance approach based on the tenets of the Gibbs−Thomson rule to include the contribution from the kinetic factor. This work provides simulations of the process that are found to corroborate experimental results addressing the “size versus number” question. The model reveals that agglomeration of like-handed particles is not required to rationalize either the net movement of molecules from one solid enantiomorph to the other or the exponential form of the temporal enantiomeric excess profile. This work highlights the subtle interplay between the driving forces represented by crystal size (solubility, thermodynamics) and molecule number (frequency, kinetics) in the process of the evolution of single chirality in the solid phase.

crystal size under conditions where the initial difference in average initial crystal size becomes large. The simulations corroborate the experimental trends, delineating the competing influences of crystal size and number of molecules on the direction of the homochiral outcome. The conditions under which the parameter controlling the direction of homochirality switches between crystal size and crystal ee is a function of specific model parameters, but the trends shown for the parameters used may be discussed in terms of other influences. Thus the model predicts that slowing the solution racemization rate causes the switch to dominance in number over size to occur at a lower enantiomeric excess. Decreasing the particle breaking rate also decreases the enantiomeric excess value at which number dominates over size (see Supporting Information). The role of agglomeration may be examined by including the agglomeration term of eq 9 in the mathematical model (see Supporting Information for details). Figure 10 shows that the

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ASSOCIATED CONTENT

* Supporting Information S

Experimental and analytical procedures and details of the mathematical modeling. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding from the U.K. EPSRC (to O.K.M. and D.G.B.), the Royal Society (Wolfson Research Merit Award to D.G.B.), the U.S. National Science Foundation (CBET-1066608 to D.G.B.), the U.S. NASA (Exobiology NNX12AD78G to D.G.B.), and the Simons Foundation (Simons Collaboration on the Origins of Life to D.G.B.), is gratefully acknowledged. D.G.B. thanks M. Mower for modeling of population genetics in Figure 1b.

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Figure 10. Evolution of solid phase enantiomeric excess for the case where an agglomeration term is included in the model compared to the case where clusters do not contribute.

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CONCLUSIONS Experimental studies of the uptake of molecules from concentrated solutions onto crystals of different sizes supports findings in attrition-enhanced deracemization that the sense of the emergence of homochirality are dictated by both a thermodynamic factor (the solubility considerations of the Gibbs−Thomson rule) and a kinetic factor (the total exposed crystal surface area). Either factor may dominate the outcome. The minor enantiomer present at the outset may control the outcome when its average crystal size is sufficiently larger (and its solubility sufficiently lower) than that of the enantiomer in excess. A mathematical model is developed that modifies a 936

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