Article pubs.acs.org/crystal
On the Effect of Initial Conditions in Viedma Ripening Martin Iggland, Roland Müller, and Marco Mazzotti* Institute of Process Engineering, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland ABSTRACT: The chiral separation process known as Viedma ripening, or attritionenhanced deracemization, is a crystallization-based process that can convert a racemic solid phase to an enantiomerically pure one. This process relies on several interacting mechanisms and is influenced by thermodynamic and kinetic factors as well as by the initial conditions of the experiment, sometimes making the interpretation of experimental results difficult. In this work, we use the mathematical model of the process that we have developed previously to explain the large variations in terms of outcome and process time observed in many experiments and to discuss them in detail in terms of the effect of the initial conditions. We show that the direction of evolution can be predicted by the model and depends on the asymmetries present in the initial particle populations. The process time is shown to be very sensitive to the initial conditions when the system is not clearly biased toward one enantiomer. This analysis shows that the lack of reproducibility of some Viedma ripening experiments can at least partially be explained by the high sensitivity of the process to the initial conditions and can be prevented by a careful choice of the initial conditions. Finally, our results are discussed in light of the practical application of Viedma ripening.
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INTRODUCTION Viedma ripening, or attrition-enhanced deracemization, is a crystallization-based process in which a racemic, or nearly racemic, solid phase is converted to an enantiomerically pure solid phase. This process was first shown to work by grinding a suspension of chiral crystals of an achiral solute.1 Later, it was shown that the process works for chiral molecules that can be racemized in solution.2 Since then, alternative methods have been proposed, most recently the use of temperature cycles3 as well as a combination of temperature and pressure cycles using high-pressure homogenization.4 In all cases, the enantiomeric excess of the solid phase increases exponentially toward an optically pure state. The mechanisms leading to the observed behavior are believed to be size-dependent growth and dissolution, racemization in solution, attrition, and agglomeration.1,5−8 We have previously published a population balance equation (PBE) model that includes all these mechanisms.9 A detailed analysis of the influence of the various parameters led to the conclusion that the necessary mechanisms are growth and dissolution, racemization, and agglomeration, while attrition, though not being necessary, greatly accelerates the process. The recently published Monte Carlo model of Ricci et al.10 comes to the same conclusion, namely that agglomeration is necessary for Viedma ripening. It is worth noting that there is also an alternative mechanism that allows for the purification of a mixture of enantiomers into an optically pure solid phase without requiring agglomeration; its behavior is different from that of Viedma ripening, and the alternative process requires that several special conditions be fulfilled.11 We note in passing that in a recent paper,12 Blackmond and co-workers argue in favor of a PBE model with breakage but without agglomeration and claim that the model can © 2014 American Chemical Society
qualitatively describe the Viedma ripening process. Unfortunately, their model is incorrect, and so is the authors’ conclusion that agglomeration is not necessary. In fact, they use a physically incorrect growth rate expression, where the growth rate of a single particle increases not only with supersaturation, as it must, but also with the concentration of crystals, which is physically inconsistent. The interested readers can convince themselves by applying Blackmond’s growth model to a simple seeded growth process at constant supersaturation and observe that during growth the initial population of seeds becomes distorted in an unphysical way. In their paper, Ricci et al.10 pointed out again the importance of understanding the influence of the initial conditions, as already highlighted by experimental observations. A characteristic of Viedma ripening experiments is in fact that the direction of evolution seems to be random when the process is started without an initial asymmetry,1 although some authors report a bias toward one of the two enantiomers.13,14 Additionally, some authors have reported a lack of reproducibility of their experiments.15,16 It is generally accepted that the direction of evolution of the system can be influenced by biasing the initial conditions toward the desired enantiomer, for example, by imposing an initial enantiomeric excess. The enantiomeric excess is a measure of the difference in mass between the two chiral particle populations and does not take into account any differences in the size of crystals. A suspension with a solid phase enantiomeric excess of 0% can still be asymmetric because of a difference in the size distribution of the two populations, i.e., in the average size or in the variance of the Received: February 6, 2014 Revised: March 20, 2014 Published: April 8, 2014 2488
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Table 1. Summary of the Model Equations That Form the Basis for Our Model of Viedma Ripening (ref 9)a
a
The equations represent the population balance equations (1), the mass balances (2), the growth rate (3), the agglomeration rate (4), and the daughter distribution (5). Nomenclature: y, particle size; τ, dimensionless time; f i, particle size distributions; Gi, growth rates; A, agglomeration rate; α, capillary length; y1 and y2, sizes of agglomerating particles; Si, supersaturations; φi,3, third moment of PSDs; g, daughter distribution. The meaning and values of the parameters in the equations are given in Table 2.
daughter distribution to describe the statistics of the outcome of breakage events. The initial populations are assumed to be normal distributions. These are described by the third moment, which is proportional to the crystal mass, the mean particle size, and the standard deviation, or width, of the distribution. In the following, various simulations are presented in which the two initial particle populations have different initial conditions. The parameters used in all simulations, unless stated explicitly otherwise, are listed in Table 2. These parameters (where Kbg =
distributions. This is most likely to occur when the two enantiomers come from different sources. There can of course be a difference between the two populations in terms of both mass and size such as in the case described by Kaptein et al.,17 who mixed ground particles of the R enantiomer with larger particles of the S enantiomer in proportions that led to an enantiomeric excess of the R enantiomer; in their experiment, the R enantiomer was the one present at the end of the process. In such cases, where the system is biased in opposing directions by different factors inducing asymmetry, the direction of evolution is not immediately clear. Several other authors have proposed that there is competition between the two and that the population with either the larger particles or the higher mass (i.e. more particles) will remain at the end of the process.10,13,15 In our previous work, we hypothesized that there is a critical threshold in terms of initial conditions and system parameters at which the outcome switches from one form to the other.9 In this context, our aim here is to investigate the effect of various initial conditions using our previously published population balance model.9 This is achieved by varying parameters describing the two initial particle size distributions, such as the particle mass for each enantiomer (leading to different values of the enantiomeric excess), the mean particle size, and the width of the particle size distribution.
Table 2. Parameters Used for All Simulations in This Work (unless explicitly mentioned otherwise)a breakage rate constant, Kbg holdup ratio, ν particle size scaling factor, x0 racemization rate constant, Krg agglomeration rate parameter, r1 agglomeration rate parameter, r2 (m3) daughter distribution parameter, q initial supersaturation, S0i (i = D or L) initial PSD standard deviation, σ0i (i = D or L) initial third moment of the D population, φ03,D a
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MODEL The model we use is based on population balance equations and takes into account the effects of size-dependent growth, attrition, and agglomeration on the two enantiomerically pure populations of crystals. Mass balances account for the solution phase, in which the racemization reaction takes place. The model has been discussed in detail previously,9 and the main equations are summarized in Table 1. It is important to note that while our model is not stochastic, it does account for statistical variations in the included mechanisms. Population balance models account for a large number of particles of different sizes by default. The rate parameters should be viewed as average values, and the fact that individual breakage events can have different outcomes is accounted for by the use of a
1, 10, 25, 100 1 1 × 10−5 10 1 × 1010 1 × 10−5 10 1 0.1 5 × 1013
Except for r2, all parameters are dimensionless.
10) correspond to the base case simulation described in our previous work. The model is made dimensionless using the kinetic rate constant of the growth rate expression and the mean initial particle size, as described in detail in the original work. Because the model is dimensionless, the results presented here are general and not specific to a certain system. For a discussion about the generality of the rate parameters and the constitutive equations describing the exact modalities of the mechanisms, the interested reader is referred to the section entitled Discussion of Assumptions in our previous work.9 The equations are solved using the same numerical method as described previously.9 2489
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which the enantiomeric excess reaches 90% (or −90%). For this set of simulations, the τ90 values range from ∼400 to >900, with a mean value of 560 and a standard deviation of ∼90 (all in dimensionless units). With a substantial initial bias toward enantiomer D, on the other hand, the observed variation between individual simulations becomes much smaller. In particular, the profiles of the enantiomeric excess in 100 simulations are much more similar, as shown in Figure 1b, despite the random variation of the initial conditions around the reference initial state (ee0D = 5.5%, or ee0D = 25%). In this case, the mean τ90 values are around 285 and 140, and the standard deviations are only ∼10 and ∼2, respectively, as shown in Table 3.
RESULTS Randomized Initial Conditions. We have conducted three sets of simulations, which we will term sets A−C, consisting of 100 simulations each, within which the initial conditions vary slightly. In all sets, the mean particle size of the crystals varies around a value of 1, with a standard deviation of 0.005, while the mean particle size of D crystals is fixed at 1; the width of both populations varies around 0.1, with a standard deviation of 5 × 10−4, corresponding to 0.5%. The enantiomeric excess of the D enantiomer varies around 0% in set A, around 5.5% in set B, and around 25% in set C, with a standard deviation of 0.005 in each case (we report ee values in terms of D, but this is arbitrary because the two values are related by the equation eeD = −eeL). In other words, set A is centered around the perfectly symmetrical state, while sets B and C are centered around a state with a substantial initial bias toward D. For all three sets, the breakage rate constant (Kbg) equals 10. These values were chosen to reflect the approximate variations that can reasonably be expected in a large set of experiments; these will naturally occur, even if the operators conduct the experiments very carefully trying to repeat the initial conditions with great accuracy. The overall evolution of the solid phase enantiomeric excess during these simulations, as seen in Figure 1, exhibits very
Table 3. Summary of Deracemization Times for Simulation Sets A−C Starting around Different Reference ee0D Valuesa set
reference ee0D (%)
A B C
0 5.5 25
τ9̅ 0 560 285 140
standard deviation of στ 90 10 2
a τ9̅ 0 represents the mean τ90 value, and στ is the standard deviation of the τ90 values. τ90 is the time required to reach a 90% enantiomeric excess.
Both the deracemization time and the variation decrease as the breakage rate constant is increased. Figure 2 shows a
Figure 2. Profiles of enantiomeric excesses for 100 simulations with initial conditions with small initial biases. The black curves are the same ones shown in Figure 1a (Kbg = 10), while the other two sets are for higher breakage rate constants (red for Kbg = 25, and blue for Kbg = 100).
comparison of three sets of simulations with breakage rates (Kbg) of 10 (set A, shown above), 25, and 100. As the breakage rate constant increases, the average deracemization time decreases from 560 at a Kbg of 10 to 300 at a Kbg of 25 and to 120 at a Kbg of 100, thus following the expected behavior.9 Furthermore, the standard deviation decreases from 90 at a Kbg of 10 to 60 at a Kbg of 25 and to 25 at a Kbg of 100. Ricci et al.10 have used Monte Carlo simulations to conduct a similar study of the effect of initial conditions, namely also starting without an initial bias (ee0D = 0%) or with an enantiomeric excess ee0D of 5.5%. A very important difference to the simulations presented here is that they start at exactly the same conditions for all simulations, but the Monte Carlo model naturally includes stochastic effects during the process. Nevertheless, the results are similar to those presented here: the differences between simulations are larger when starting without an initial bias than when starting with a bias. Note that although the differences observed when starting with an ee0D of
Figure 1. Profiles of the solid phase enantiomeric excess for 100 simulations in each set with initial conditions with (a) small and (b) large initial biases.
different behavior in the two figures. For the simulations of set A, without a substantial initial bias toward a specific enantiomer, the system evolves toward a state pure in either the D or the L enantiomer depending on the initial asymmetry with a widely varying rate, as shown in Figure 1a. A measure of the rate of the process is the τ90 value, which is the time at 2490
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In the following sections, we will show why the initial conditions have such a strong influence on the process and how an understanding of this influence can help in the design more robust Viedma ripening experiments and processes. Direction of Evolution. First, we analyze the direction of evolution of the system as a function of the initial conditions. For this, we consider the parameters characterizing the initial particle size distributions: the initial enantiomeric excess, the initial mean particle sizes, and the initial standard deviations. A system is completely symmetric if the solution phase concentration for both enantiomers is equal and the particle size distributions of the two populations of particles are identical. In our case, for normally distributed particle sizes and a saturated solution, this means that perfect symmetry occurs if the solid phase enantiomeric excess ee0D is 0 and the mean particle sizes and the standard deviations are equal; i.e., y0D̅ = y0L̅ , and σ0D = σ0L . These conditions correspond to the point at which ee0D, y0L̅ /y0D̅ = 1, σ0L /σ0D = 1 in a three-dimensional (3D) parameter space spanned by the coordinates ee0D, y0L̅ /y0D̅ and σ0L /σ0D; two two-dimensional subspaces of the 3D space are shown in Figure 4. At the symmetry point, there is no imbalance and thus no evolution of the enantiomeric excess. However, the slightest deviation from this point yields an asymmetry that is sufficient to trigger the evolution of the system toward complete deracemization, i.e., chiral purity. By running simulations with varying initial conditions (as marked by the points in Figure 4), we have created a map of
5.5% appear to be larger in the work of Ricci et al. than in Figure 1, the results cannot be quantitatively compared because their model and our model use different parameters. Recently, Hein et al.15 have reported experiments that were started with racemic material, i.e., without an intentional initial bias. The profiles of the enantiomeric excess look remarkably similar to those shown in Figure 1a; a large part of the variation in their measured profiles is attributed by the authors to random events during the process. Hein et al. monitored the particle size during two experiments that proceeded to complete deracemization relatively quickly, and during one experiment in which there seemed to be no progress (i.e., there was only a slow increase in the enantiomeric excess). They observed that the particle size increased near the time when the system reached chiral purity, whereas in the experiment that did not proceed toward deracemization, there was no increase in particle size. Their hypothesis is that “larger crystals are stochastically formed”,15 thus triggering the start of the deracemization process. In our simulations, we observe a similar behavior: the mean particle size increases, but this increase accelerates only toward the end of the deracemization process. The profiles of the mean particle size of both individual particle populations as well as the overall mean particle size are shown together with the profile of the enantiomeric excess in Figure 3 for one of the simulations of set A; all the other
Figure 3. Profiles of enantiomeric excesses (top) and mean particle sizes (bottom; blue for yD̅ , red for yL̅ , and black for overall) for one of the simulations in set A.
simulations exhibit similar behavior. The mean size of the crystals of the minor enantiomer decreases due to net dissolution, while the mean size of the crystals of the major enantiomer increases due to net growth and agglomeration; once the minor enantiomer has been depleted sufficiently, the size increase is visible in the overall mean particle size, as well. This occurs only once the enantiomeric excess has reached a certain level, which is exactly what Hein et al. observed in their experiments. The variation in the average particle size after completion of deracemization observed by Hein et al. is also exhibited in our simulations, to an extent that depends on the values of the model parameters. The kink in the curve for the L enantiomer around a τ of 400 is due to the fact that we assume breakage events result in two fragments, one small and one large, thus yielding a bimodal distribution. The smaller particles dissolve faster and are mostly completely dissolved near a τ of 400. Thereafter, the decrease in the average particle size decelerates temporarily because the remaining particles are larger and therefore dissolve slower. It is worth pointing out that the outcome of our simulations does not depend on any random effect during the process, but only on the random initial asymmetry.
Figure 4. Parameter plane spanned by the initial solid phase enantiomeric excess and (a) the initial ratio of mean particle sizes or (b) the initial ratio of the standard deviations of the two populations. Individual simulations are shown at their initial point, with the marker showing the final state. In both cases, two regions are visible, one which results in a final state containing only D crystals (blue triangles) and the other of which results in a final state with only L crystals (red dots). The estimated separatrix between these two regions is shown as a solid black line. 2491
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of the particles in the suspension. The separatrix will asymptotically move toward a vertical line at ee0D = 0 for increasing breakage rate constants; i.e., the size difference will become less important, and the outcome will be determined mainly by deviation of the initial enantiomeric excess from zero. It should be noted that the mean particle size of the D crystals was kept constant, and thus, there is no reason to expect symmetry above and below the y0L̅ /y0D̅ axis. The overall particle size is smaller below the axis than above it, and thus, the shape of the curve is also different. Deracemization Time. The deracemization time τ90 of three sets of simulations with equal values for the initial standard deviations and different values for the ratio of the initial mean particle sizes (y0L̅ /y0D̅ ) is shown as a function of the initial enantiomeric excess in Figure 6. These correspond to
outcomes in both parameter planes described above. The red dots in the figure mark simulations that end up with only L crystals remaining, while the blue triangles mark simulations ending with only D crystals. Along the horizontal axes y0L̅ /y0D̅ = 1 and σL0/σ0D = 1 in Figure 4, the enantiomeric excess is controlling, and the enantiomer that is initially in excess prevails. When there is no initial enantiomeric excess, i.e., along the line ee0D = 0 (the vertical axis) in Figure 4, the particle size is controlling when the initial standard deviation is the same (Figure 4a, where the enantiomer whose initial mean particle size is largest prevails), and the initial standard deviation is controlling when the initial average size is the same (Figure 4b, where the enantiomer whose initial distribution is wider prevails). Both the average size and the width of the particle size distribution affect the growth and dissolution of particles caused by the size dependence of solubility.18 Smaller particles dissolve preferentially, while a large size variation means that the driving force for growth is larger and thus net growth will start earlier than for a narrower particle size distribution. The influence of the initial mean particle size is more pronounced than that of the size variation (note the scale of the ee0D axis in Figure 4b). Thus, in the top left quadrant of both figures, two parameters are biased toward L crystals and thus the outcome is clearly L. The opposite is true for the bottom right quadrant, where both parameters are biased toward the D enantiomer. On the other hand, in the bottom left and top right quadrants, two imbalances act in opposite directions. As shown by simulations, the consequence is that a separatrix divides these quadrants into two sectors. Along the separatrix, the two effects counterbalance and in principle no deracemization occurs, whereas on the two sides of it, deracemization proceeds in the direction dictated by the parameter that has the stronger effect in that sector. Of course, this separatrix passes through the point of symmetry at the origin of the 3D parameter space. This division of the two planes in panels a and b of Figure 4 into two sectors by a smooth line can be extrapolated into 3D space, where the two sectors are divided by a smooth surface. The system deracemizes toward pure enantiomer D in the sector containing the portion of the ee0D axis where ee0D > 0, the portion of the y0L̅ /y0D̅ axis below 1, and the portion of the σ0L /σ0D axis below 1, and toward pure enantiomer L in the other sector. The shape of the separatrix depends on the thermodynamics and kinetics of the process. Figure 5 shows the separatrix in the ee0D versus y0L̅ /y0D̅ plane (as in Figure 4a) for three different breakage rate constants. More intense breakage thus weakens the influence of the difference in particle size on the outcome. This is expected, because breakage leads to a homogenization
Figure 6. Deracemization time τ90 as a function of the initial solid phase enantiomeric excess for different ratios between the initial mean particle size of the two populations. Lines are to guide the eye.
simulations along horizontal lines in Figure 4a, at vertical coordinates 0.5, 1, and 2.25. In all three series, simulations to the left of the peak end up with only the L enantiomer remaining, and with only the D enantiomer remaining to the right of the peak. The peak occurs at the position corresponding to the separatrix discussed above, and it is clear that the deracemization time is very sensitive to the exact initial conditions near this separatrix, which is characteristic of the process, and that the time increases toward infinity, i.e., no deracemization, the closer to the separatrix the system moves. Because Viedma ripening is an autocatalytic process, deracemization is faster the more asymmetric the system is. Thus, the farther one moves away from the separatrix, the faster the deracemization and the less sensitive the process is to the initial conditions, as evidenced by the plots in Figure 6. Interestingly, the contour lines of the surface of τ90 values follow the separatrix, as shown in Figure 7. This implies that the main factor determining the deracemization time is the distance of the initial conditions from the separatrix.
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DISCUSSION AND CONCLUSIONS Viedma ripening experiments reported in the literature have sometimes suffered from a lack of reproducibility, making their interpretation difficult. As our simulations show, experiments starting with racemic material are especially susceptible to variations caused by small differences in the initial conditions, and these small differences can explain the observed behavior, even without any stochastic events occurring during processing. Such differences are inevitable in experiments and depend on many factors. Even a racemic mixture obtained from a single
Figure 5. Boundary curves for three different breakage rate constants (Kbg) of 1 (dotted), 10 (solid), and 100 (dashed). 2492
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work provides valuable information to those who might want to exploit Viedma ripening for preparative purposes.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Viedma, C. Phys. Rev. Lett. 2005, 94, 065504. (2) 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, 1158−1159. (3) Suwannasang, K.; Flood, A. E.; Rougeot, C.; Coquerel, G. Cryst. Growth Des. 2013, 13, 3498−3504. (4) Iggland, M.; Fernández-Ronco, M. P.; Senn, R.; Kluge, J.; Mazzotti, M. Chem. Eng. Sci. 2014, 111, 106−111. (5) Uwaha, M. J. Phys. Soc. Jpn. 2004, 73, 2601−2603. (6) Noorduin, W. L.; Meekes, H.; Bode, A. A. C.; van Enckevort, W. J. P.; Kaptein, B.; Kellogg, R. M.; Vlieg, E. Cryst. Growth Des. 2008, 8, 1675−1681. (7) McBride, J. M.; Tully, J. C. Nature 2008, 452, 161−162. (8) Noorduin, W. L.; van Enckevort, W. J. P.; Meekes, H.; Kaptein, B.; Kellogg, R. M.; Tully, J. C.; McBride, J. M.; Vlieg, E. Angew. Chem., Int. Ed. 2010, 49, 8435−8438. (9) Iggland, M.; Mazzotti, M. Cryst. Growth Des. 2011, 11, 4611− 4622. (10) Ricci, F.; Stillinger, F. H.; Debenedetti, P. G. J. Chem. Phys. 2013, 139, 174503. (11) Iggland, M.; Mazzotti, M. CrystEngComm 2013, 15, 2319−2328. (12) Gherase, D.; Conroy, D.; Matar, O. K.; Blackmond, D. G. Cryst. Growth Des. 2014, 14, 928−937. (13) Noorduin, W. L.; Vlieg, E.; Kellogg, R. M.; Kaptein, B. Angew. Chem., Int. Ed. 2009, 48, 9600−9606. (14) Steendam, R. R. E.; Harmsen, B.; Meekes, H.; Enckevort, W. J. P. v.; Kaptein, B.; Kellogg, R. M.; Raap, J.; Rutjes, F. P. J. T.; Vlieg, E. Cryst. Growth Des. 2013, 13, 4776−4780. (15) Hein, J. E.; Huynh Cao, B.; Viedma, C.; Kellogg, R. M.; Blackmond, D. G. J. Am. Chem. Soc. 2012, 134, 12629−12636. (16) Rougeot, C. Deracemisation of active compound precursors by physical treatments. Ph.D. Thesis, University Paul Sabatier of Toulouse III, Toulouse, France, 2012. (17) Kaptein, B.; Noorduin, W. L.; Meekes, H.; van Enckevort, W. J. P.; Kellogg, R. M.; Vlieg, E. Angew. Chem., Int. Ed. 2008, 47, 7226− 7229. (18) Iggland, M.; Mazzotti, M. Cryst. Growth Des. 2012, 12, 1489− 1500. (19) Siegel, J. S. Chirality 1998, 10, 24−27.
Figure 7. Contour plot of τ90 values as function of initial solid phase enantiomeric excess and ratio of initial mean particle sizes. The thick black line is the separatrix shown in Figure 4a.
crystallization step is unlikely to contain exactly equal numbers of particles of both enantiomers;19 if the mixture is created by mixing two chirally pure solids, the differences between experiments can be even larger. The fact that initial conditions can vary between experiments, and that these can have a drastic effect on the outcome of the experiments, needs to be considered when designing and evaluating experiments and also when analyzing the results of simulations. To obtain reliable results that can provide information about the influence of various parameters on the kinetics of the process, it is important that the results are not skewed by other effects, and thus, it can be better to start with an initial bias of a few percent, say 5−10%, in the enantiomeric excess when running Viedma ripening experiments. In a production setting, it is important not only that the correct enantiomer is obtained but also that the process time is predictable and not too long. Most likely, the material to be deracemized will be obtained by a synthesis step resulting in a racemic mixture. If this material is crystallized, there will not be large differences between the particle size distributions of the two enantiomers. The easiest way to push the process in the desired direction and to reduce both the process time and its variation is to add pure seeds obtained in a previous run, thus increasing the initial ee above 0%. Additionally, the size of these seeds may well be different from that of the new material; if they are larger, they will also lead to a biasing toward the desired enantiomer. As shown by our simulations and summarized in Table 3, the more seeds are added, the shorter the process time and the smaller the variation in process times, but of course, more material is required. As the initial enantiomeric excess increases, the incremental decrease in the deracemization time becomes smaller and smaller, and thus, the benefit of adding extra material will become negligible at some point. When exactly this occurs is of course dependent on both the system and the process specifications chosen by the operator, but we expect that it will be in the range of 5−10%. This work demonstrates how the PBE model developed recently9 and used here is able not only to elucidate the complex mechanisms of Viedma ripening but also to clarify the effect that varying initial conditions and random initial asymmetries have on the outcome and speed of deracemization. Random initial asymmetries are inevitably present in lab experiments, and our analysis rationalizes some of the experimental observations reported in the literature. Intentionally biased initial conditions allow directing deracemization toward the target enantiomer and controlling its speed, and this 2493
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