Linear deracemization kinetics during Viedma ripening: Autocatalysis overruled by chiral additives

Article abstract of DOI:10.1021/acs.cgd.5b00127

Viedma ripening proceeds through an autocatalytic feedback mechanism which exponentially deracemizes an initially racemic solid state to an enantiopure end state. Here we show that, in the presence of enantiopure additives with a concentration of as low as 2.5 × 10^-2 mol %, Viedma ripening proceeds with an overall linear and faster increase in enantiomeric excess. These experimental results can be explained using a simple model which assumes a difference in growth and dissolution rates between the enantiomers. This model also accounts for the generally observed linearity during the initial stages of Viedma ripening without additives.

Full text of DOI:10.1021/acs.cgd.5b00127

Article
pubs.acs.org/crystal
Linear Deracemization Kinetics during Viedma Ripening:
Autocatalysis Overruled by Chiral Additives
Rene R. E. Steendam,† Janneke Dickhout,† Willem J. P. van Enckevort,*,† Hugo Meekes,† Jan Raap,‡
́
Floris P. J. T. Rutjes,† and Elias Vlieg†
†Radboud University Nijmegen, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 ED Nijmegen, The Netherlands
‡Leiden Institute of Chemistry, Leiden University, Einsteinweg 55, 2333 CC, Leiden, The Netherlands
ABSTRACT: Viedma ripening proceeds through an autocata-
lytic feedback mechanism which exponentially deracemizes an
initially racemic solid state to an enantiopure end state. Here
we show that, in the presence of enantiopure additives with a
concentration of as low as 2.5 × 10−2 mol %, Viedma ripening
proceeds with an overall linear and faster increase in enantio-
meric excess. These experimental results can be explained
using a simple model which assumes a difference in growth and
dissolution rates between the enantiomers. This model also
accounts for the generally observed linearity during the initial
stages of Viedma ripening without additives.
account for the exponential deracemization rate observed in
many Viedma ripening experiments.18,22−25
INTRODUCTION
■
Acquiring enantiopure molecules is of profound importance in
many fields of chemistry.1 Viedma ripening2,3 is among the most
facile and reliable methods available to reach single chirality.4
It enables the complete solid-state deracemization of molecules
that form racemic conglomerate crystals and racemize in
solution. To date, many compounds have been deracemized
successfully through Viedma ripening.5,6 The initial symmetry
breaking of the solid state, and with that the final configuration
of the product, depends on the initial enantiomeric excess (ee),7
the difference in crystal size distribution (CSD) between the
enantiomers,8 unintended chiral impurities,9 or chiral additives.7
Chiral additives hamper the crystallization and dissolution of
their corresponding enantiomer leading to a difference in CSD
between the enantiomers.10
In the Viedma ripening process, symmetry breaking is
followed by asymmetric amplification of the solid state ee,
which proceeds through a mechanism that can be explained by
taking four processes into account: (1) racemization in solution,
(2) Ostwald ripening, (3) attrition, and (4) enantioselective
reincorporation of chiral clusters.11 The effect of enantioselective
incorporation of chiral clusters into larger crystallites of the same
hand is required to explain the exponential increase in ee.11−15
Enantioselective agglomeration of crystals has been observed
experimentally for sodium chlorate and sodium bromate.16
However, chiral recognition at a smaller scale between clusters
and crystals has not been demonstrated yet, although achiral
clusters and their interactions have been studied.17 Nevertheless,
apart from a single report,18 all computational studies in
literature based on rate equations,12,13 Monte Carlo simula-
tions,14,19 a population balance model,20 and dispersive kinetic
models21 require a feedback mechanism in terms of clusters to
Recently, deracemization of conglomerate crystals was
accomplished through repeated temperature-induced dissolu-
tion and growth of a racemic mixture of crystals in solution.26
Differences in growth rates of crystals was proposed as the driving
force behind this deracemization method using temperature
cycling.27 These growth rate differences can be explained by the
nucleation process of the crystals prior to deracemization in which
nucleation of both enantiomers occurs at slightly different times
and thus at different supersaturation levels. These differences lead
to crystals that individually have slightly different defects and
therefore slightly different thermodynamic and kinetic properties.
Two models have been proposed in which either the mean crystal
growth rate or the growth rate dispersion was varied. Starting
without a difference in CSD and ee, each model resulted in a
sigmoidal increase in ee. As crystal breakage and agglomeration
are not considered and temperature is constant, these models27 at
present cannot be used to explain Viedma ripening.
Here we show experimentally that deracemization through
Viedma ripening in the presence of chiral additives leads to a
linear instead of the previously observed exponential increase
in ee. These results can be explained by means of a model which
assumes a difference in growth and dissolution rate between the
enantiomers.
EXPERIMENTAL SECTION
■
The herein reported experimental procedures are based on a literature
method.9 Each experiment was conducted using new glassware in
Received: January 28, 2015
Revised: March 2, 2015
© XXXX American Chemical Society
A
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order to avoid any effect of (chiral) contamination. In these experi-
ments, N-(2-methylbenzylidine) phenylglycin amide (1) was used
from the same batch as the experiments reported previously.7,9 The ee
of the starting material (1) was found to be 0% within the detection
limit (∼0.5%) of the chiral HPLC. Polytetrafluoroethylene (PTFE)-
coated oval shaped magnetic stirring bars (length 20 mm, ø 10 mm)
and scintillation flasks were purchased from VWR. Glass beads
(ø = 1.5−2.5 mm), 1,8-diazabicyclo[5.4.0]undec-7-ene (DBU), both
enantiomers of phenylglycine (2) and acetonitrile (ACN) were purchased
from Sigma-Aldrich and used as received. The experiments were
conducted at a temperature of 20 2 °C.
Viedma Ripening with Additives. A 20 mL scintillation flask was
filled with 400 mg of (rac)-1, 0.025−200 mg of (S)-2 or (R)-2, 8 g of
glass beads, an oval shaped magnetic stirring bar, and 3.5 g of ACN.
For experiments in which a small amount of additive (<1 mg) was
used, a stock solution was prepared by dissolving 1 mg of (S)-2 in
4.0 g of ACN from which the required amount of additive was taken.
The resulting suspension was ground at 600 rpm for 1 h after which
0.147 mL of DBU was added to start the deracemization process.
Sampling. About 0.2 mL of slurry was withdrawn from the experi-
ment using a pasteur pipet. The sample was immediately subjected to
vacuum filtration on a P4 glass filter (ø 10 mm). The residual crystals
were washed on the filter using 0.5 mL of methanol in order to remove
DBU and mother liquor. The crystals were subsequently dissolved in
1.5 mL of 2-propanol and the ee was determined using chiral HPLC.
Determination of ee Using Chiral HPLC. A Chiralpak AD-H
(250 × 4.6 mm ID) column was used in combination with the
following conditions; eluent n-heptane/2-propanol (95/5 v/v%), flow
1 mL/min, room temperature, λ = 254 nm, inj. vol. 20 μL. Retention
times: (R)-1 30.1 min, (S)-1 34.5 min.
Without additives, the initial racemic solid state slowly becomes
enantiomerically enriched after which the ee increases
exponentially. Adding a small amount (0.4 mol %) of additive
results in a much faster deracemization rate in which the ee was
found to increase in a linear fashion.
The linear increase was reproducibly found for different
amounts of additive for experiments conducted in acetonitrile
(ACN, Appendix A1). In a previous report, the additive amount
used varied from 0.1 to 8.7 mol %.7 The fastest deracemization
rate in our experiments was found to be in the presence of
about 1 mol % of additive (Figure 2b). Similar linear growth
kinetics was observed during Viedma ripening experiments
conducted in methanol (MeOH, Appendix A2). In this case the
deracemization rate was lower, which can be explained by
slower racemization of 1 in MeOH (Appendix B). For additive
concentrations above 80 mol % deracemization of 1 is almost
completely inhibited.
MODELING RESULTS
■
Model Description. To understand the linear behavior of
the observed ee(t) curves, we start with the simple model for
Viedma ripening as described previously,11 which is based on
the original model of Uwaha.12−15 The only difference with the
previous model is that we now allow for a difference in growth
and dissolution rates between the enantiomers. This is based
upon the experimental observation of a different size distribu-
tion between the enantiomers in the presence of enantiopure
additives during grinding.10 In this model only two crystal sizes
are taken into account: big crystals (containing in total B+ and
B− molecules of the two enantiomers) and clusters (containing
C+ and C− molecules altogether). Figure 3 schematizes the
various processes involved in the deracemization procedure,
which is described by the set of coupled differential eqs 1a−1d
with analogous equations for the opposite enantiomer.
Determination of Racemization in Solution. Ten milligrams of
(R)-1 was dissolved in 10.0 mL of either ACN or MeOH. To this was
added 0.1 mL of a solution containing the catalyst which was prepared
by adding 50 μL of DBU (16.9 mol %) in 5.0 mL of either MeOH or
ACN. From the resulting solution, 1.0 mL was transferred to an HPLC
vial, and the ee of the stagnant solution was monitored through time
using chiral HPLC analysis. The results are plotted in Figure 11 of
Appendix B.
∂B+
= a+B+(M+ − MeBq) − bB+ + cC+B+
EXPERIMENTAL RESULTS
■
(1a)
∂t
To study the effect of additives on Viedma ripening, experiments
were conducted using a model system involving an amino acid
derivative (Figure 1, compound 1).7 The enantiopure additive
∂C+
∂t
= bB+ + a+C+(M+ − MeCq) − cC+B+
(1b)
∂M+
= −a+B+(M+ − MeBq) − a+C+(M+ − MeCq)
∂t
− d(M+ − M−)
(1c)
T = B+ + C+ + B− + C− + M+ + M− = constant
(1d)
The solid phase ee is defined as
Figure 1. Schematic representation of the deracemization of compound
1 through Viedma ripening. Chiral additive (S)-2 enantiomerically
hampers the crystal growth of (S)-1 leading to different growth and
dissolution rates for (S)-1 and (R)-1, and ultimately an enantiopure
(R)-1 end state is obtained.
(B+ + C+) − (B− + C−)
ee =
(B+ + C+) + (B− + C−)
(2)
We assume an ablation rate (formation of small chiral clusters
by grinding) proportional to the number of large crystals of
each enantiomer, bB. As fractured crystals are studded with
growth steps and the advancement of steps is proportional to
supersaturation, we consider linear growth kinetics. This means
that the growth and dissolution rate of large crystals and small
clusters is proportional to the difference in actual solute
concentration and the equilibrium concentration of the solution
(S)- or (R)-phenylglycine (2) is able to enantioselectively affect
the crystallization of its corresponding enantiomer of com-
pound 1. This leads to different crystallization and dissolution
rates between the enantiomers. As a result, deracemization
proceeds to give enantiopure 1 of which the final handedness
of 1 is opposite to the handedness of the additive 2 (known as
Lahav’s rule of reversal).28
in contact with the large crystals and small clusters, a (M −
B/C
The evolution of ee during Viedma ripening in the absence
and presence of chiral additives is shown in Figure 2a.
M
). As a consequence of the Gibbs−Thomson effect, the
eq
equilibrium concentration of the solution in contact with the
B
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Figure 2. Viedma ripening experiments conducted in ACN. (a) In the absence of additives, the ee of 1 is amplified in a sigmoidal fashion, whereas a
linear increase in ee is observed when chiral additive 2 was used. The lines are a guide to the eye. (b) Initial rate, dee/dt, of deracemization as a
function of the amount of additive.
a
Table 1. Variables and Constants Used in This Study
B+, B−
C+, C−
Number of molecules in large crystals of the + and − enantiomer
Number of molecules in small crystal clusters of the + and −
enantiomer
M
Total number of monomer molecules in solution
M+, M− Number of monomer molecules of the + and − molecules in
solution
t
Time
MBeq
Equilibrium number of molecules in solution in contact with the
big crystals of each enantiomer
MCeq
Equilibrium number of molecules in solution when in contact with
the small clusters of each enantiomer
T
Total number of molecules in the solid phase and solution
Enantiomeric excess in the solid phase
ee
a+, a−
Kinetic coefficients for growth and dissolution of the + and −
enantiomer
b
Rate constant for ablation of big crystals
Figure 3. Schematic view of the processes involved during Viedma
ripening.
c, c+, c− Rate constants for the incorporation of chiral clusters into big
crystals
d
k
Rate constant for racemization in solution
small clusters, MCeq, is larger than that of the large crystals MeBq.
The incorporation rate of small clusters into the large crystals of
the same handedness is second order, proportional to both the
number of large crystals and the number of small clusters, i.e.,
cCB. We assume no incorporation of small clusters into large
crystals of opposite handedness. Finally, the racemization rate
constant in solution is defined by d, where we assume that
the racemization rate is proportional to the difference in con-
centration of both enantiomers, d(M+ − M−). All the parameters
used are listed in Table 1.
Rate constant for deracemization at start; [∂ee(t)/∂t]t≈0 = k
a
All numbers and rates have relative meaning only.
shaped ee(t) curves as was also observed in many experiments.22−25
In our model (i.e., eqs 1a−1d) this situation corresponds to
a+ = a− and c ≠ 0 (Figure 4, solid line). Chiral additives or chiral
impurities reduce the growth and dissolution rate of chiral
crystals of the same handedness.7,9,10 Therefore, adding chiral
additives to the system leads to slightly different values of the
kinetic coefficients a+ and a−. Simulations using a fixed c ≠ 0
show that upon introducing a difference between a+ and a‑ leads
to a change in curve shape which becomes more and more linear
as shown in Figure 4. The additive effect almost completely
overrules the cluster effect provided that the additive effect
is sufficiently large (a+ − a− = 0.50, Figure 4). In addition,
deracemization proceeds much faster when enantiopure
additives are present.
To follow the process of deracemization in time, the set of
eqs 1a−1d is numerically integrated using the finite difference
method. Similar to the experiments, the system was “premixed”
for some period prior to the ee(t) runs, keeping a+ = a−
=
(a+ + a−)/2, not allowing racemization in the liquid and starting
from B+ = B− and C+ = C− = 0. This gives ee(t = 0) = 0, an
equal size distribution of both enantiomers (i.e., B+ = B− = B0
and C + = C− = C0) and avoids initial fluctuations in ee(t). In all
simulation runs we kept MBeq = 1.0 and MeCq = 1.1.
No Clusters Required for Complete Deracemization.
We find that there is no need for cluster incorporation to
achieve full deracemization by grinding, provided that a+ ≠ a−.
Solving the set of coupled differential equations with c = 0 and
∂#/∂t = 0 gives four steady state solutions at t = ∞. This follows
Simulations. In earlier work, Viedma ripening in the
absence of chiral additives was explained by the reincorporation
of tiny chiral clusters (c ≠ 0) produced by grinding, into the
larger crystals of the same handedness.11 This leads to sigmoid
C
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k = ee(t)/t (see Appendix C). For c = 0 and fast racemization,
i.e., d → ∞, this gives
k ≈ b(a+ − a−)/(a+ + a−)
(4)
This demonstrates that the kinetics are indeed linear.
Equation 4 further shows that in this approximation k
practically only depends on the grinding rate b and the kinetic
coefficients a+ and a−, and is almost independent of the total
amount of chiral molecules, and the equilibrium concentrations.
Figure 5a shows a simulated ee(t) curve (dashed line) for c = 0,
ee(t = 0) = 0 and d = ∞ and the comparison with the linear
relation ee(t) = kt (solid line), with k given by eq 4. For
low values of t an excellent fit is obtained. To validate this
expression for different parameters, a successful comparison has
been made between k values obtained from simulations and
calculated k using eq 4 (Figure 5b).
Additive Versus Cluster Effect. Both the additive and the
cluster effect can act simultaneously (c ≠ 0 and a+ ≠ a−).
Numerous simulations using a variety of parameters and
instantaneous racemization in solution indeed show in every
case that eq 4 is approximately satisfied. Figure 6a displays k
versus b((a+ − a−)/(a+ + a−)) curves for three values of c and
shows that the linear curves coincide for all c. In other words, k
is independent of c and the difference between a+ and a− gives
the ee(t) curve an initial “boost” with slope k. For further time t,
the ee(t) curve turns sigmoidal and the cluster effect determines
the deracemization process as shown in Figure 6b.
Chiral additives or impurities may also affect the incorpo-
ration of the clusters into the larger crystals, resulting in
different clustering rates (i.e., c+ ≠ c−). Figure 7a shows an ee(t)
curve for c+ ≠ c−, ee(t = 0) = 0 and a+ = a−. Again k ≠ 0, but
also sigmoidal characteristics can be observed. In this case the
initial deracemization rate is found to depend almost linearly on
c+ − c− (Figure 7b). A complete expression for k is beyond the
scope of our study. A combination of a+ ≠ a− and c+ ≠ c− gives
sigmoidal curves and again with k ≠ 0.
Effect of Racemization Rate. To verify the effect of
slower racemization rates in solution, simulation runs have been
performed for a range of d values. For all values of d the
simulation runs with a+ ≠ a− and c = 0 again produce linear
ee(t) curves except for large t as shown in Figure 8. Thus, the
initial linearity persists independently of racemization rate.
This corresponds to our linear ee(t) curves of Viedma ripening
experiments conducted in both ACN (fast racemization) and
MeOH (slow racemization). The deracemization rate constant, k,
Figure 4. Simulations of Viedma ripening experiments as a function
of the amount of additive, corresponding to a specific difference in
(a+ − a−) for fixed c. Simulation parameters: ee(t = 0) = 0.001; c =
0.01; a− = 5.00; d = 10.
from solving eqs 3a−3c, eq 1d and the analogous equations for
the opposite enantiomer.
a+B+(M+ − MeBq) − bB+ = 0
(3a)
a+C+(M+ − MeCq) + bB+ = 0
(3b)
−a+B+(M+ − MeBq) − a+C+(M+ − MeCq)
− d(M+ − M−) = 0
(3c)
The four solutions are
+
(i) B− = 0, C− = 0, with B+ + C = T - M and ee = 1
(ii) B+ = 0, C+ = 0, with B− + C = T - M and ee = −1
−
(iii) B− and B+ ≠ 0, this solution is only possible if a+ = a‑
(iv) B− = B+ = C− = C + = 0, if all crystals are dissolved in M
by too intense grinding (i.e., b is too large).
In all four cases the solution gives M+ = M− = 0.5M,
regardless of d > 0. Only solutions (i) and (ii) are relevant and
show that at t = ∞, enantiopure material is obtained by Viedma
ripening without the need for cluster incorporation (i.e., it is
possible to have c = 0) into the big crystals in the case a+ ≠ a−.
Deracemization Kinetics. The set of differential eqs 1a−1d
cannot be solved exactly, but needs approximations to derive
an analytical expression for the deracemization rate constant,
Figure 5. (a) Simulated ee(t) (dashed line) and a fit with ee(t) = kt (solid line) with a+ = 5.05; a− = 5.00; b = 0.2; c = 0. (b) k/b obtained from
simulated ee(t) slopes at t = 0 as a function of (a+ − a−)/(a+ + a‑) for different values of a+ and b, keeping a− = 10.0, c = 0 and d = ∞. If eq 4 is
perfectly satisfied, the slope of the curve is 1.0, which holds well for the smaller values of (a+ − a−)/(a+ + a−) and b.
D
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Figure 6. (a) k versus b((a+ − a−)/(a+ + a−)) for three values of c, showing that the initial slope of the ee(t) curve is solely determined by a+, a− ,and
b, and not by c. (b) After an initial linear increase ((a+ − a−) = 0.10), the cluster effect (c = 0.008) takes over, resulting in a sigmoidal increase in ee
for larger ee(t). Simulation parameters: ee(t = 0) = 0.0; a− = 5.00; b = 0.2; d = 500; MeCq = 1.1; MeBq = 1.0.
Figure 7. Different cluster incorporation rates c+ ≠ c− and a+ = a−: (a) ee(t) curve showing that k ≠ 0; (b) Linear dependence of k on (c+ − c−).
In (a) (a = 5.0; c+ = 0.015; c− = 0.01); in (b) (a = 5.0; b = 0.2; c− = 0.01).
Figure 8. Effect of slower racemization in the solution: (a) regardless of the racemization rate, the onset of the ee(t) curve is always linear; (b) k as a
function of racemization rate d in the solution. Simulation parameters: a+ = 5.05; a− = 5.00; b = 0.2; c = 0. In (a), d = 10.
decreases for decreasing d, as expected (Figure 8b). But the
linearity of ee(t) for initial t persists.
leads to a difference in the growth and dissolution rates between
the enantiomers (i.e., the difference between a+ and a−), which
completely overrules the effect of cluster incorporation.
Both compound 1 and additive 2 were used previously,7 and
in that work the ee(t) curves showed both linear and sigmoidal
characteristics similar to Figure 4. The main difference be-
tween both series of experiments is the deracemization time.
DISCUSSION OF EXPERIMENTS AND SIMULATIONS
■
Both our experiments and our model show linear ee(t) curves.
For the experiments the linearity persists almost to ee(t) = 1,
with a large slope k. This indicates that the enantiopure additive
E
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The downscaling in our experiments results in a 1−2 orders of
magnitude larger grinding rate (i.e., b in our model) that leads
to a 50 times faster deracemization rate as compared to the
previous study.7 This in turn leads to a large k-value, overruling
the effect of cluster incorporation, giving almost completely
linear ee(t) curves in contrast to the previous report.7
For very high additive concentrations k does not increase
anymore in our experiments (Appendix A). Possibly the surface
of the crystals is completely covered by the additive of the same
handedness, so a “saturation” of (a+ − a−) occurs. It is also
possible that a high chiral additive concentration hinders the
growth rate of the opposite enantiomer as well.
Following the simplest model, the initial slope of ee(t) is k ≈
b((a+ − a−)/(a+ + a−)). This implies that the effect of chiral
additives on Viedma ripening is largest if the additive induces a
difference in growth rates between both enantiomers, (a+ − a−)
and the grinding rate (b) is large. However, one cannot exclude
additive adsorption effects on the incorporation rate of clusters.
Also the rate of solution racemization plays a role. Both affect
the time development of the solid state ee, but in all cases the
ee(t) curve is linear for not too large t and its initial slope is
larger than zero.
Recently, another model for Viedma ripening has been
proposed, which is based on crystal size dependent growth rates
by introducing a frequency term in the crystal growth rate
expressions.18 This gives comparable sigmoid ee(t) curves as
observed for the cluster model. We expect that also for these
approaches a difference in growth rates, a+ ≠ a−, gives the ee(t)
curves an initial “boost” with slope k. This will be verified in a
forthcoming paper.
Figure 9. Deracemization experiments conducted in ACN. Linear
increase in ee for different amounts of additive. The rate of
deracemization as a function of the amount of additive calculated
from linear fits of the data is shown in Figure 2b in the main text.
MeOH (420 min). As a result, the deracemization rate in
MeOH is much smaller than in ACN, and a larger amount of
additive is required to significantly increase the deracemization
rate (Figure 10). About 20 mol % of additive was found to
maximally increase the ee in Viedma ripening experiments
conducted in MeOH. A larger amount of additive leads to a
substantial decrease in the deracemization rate.
APPENDIX B: SOLUTION PHASE RACEMIZATION
Figure 11 shows the experimental results of solution phase
racemization of compound 1 in both ACN and MeOH.
■
CONCLUSIONS
■
Deracemization through Viedma ripening was experimentally
found to proceed in a linear fashion when enantiopure additives
are present. This can be explained by taking a difference in
growth and dissolution rates between the enantiomers into
account. The model gives a linear onset of the solid state ee(t)
curve. A competition between the additive effect and the effect
of chiral cluster incorporation into the enantiomer of the same
handedness will take place if time progresses. If the additive
effect is dominant, then the ee(t) curve remains linear for
large t and deracemization proceeds fast. If the effect of the
additive is less strong, then the cluster incorporation effect
(or another sigmoidal ee(t) curve forming phenomenon) will
take over and the ee(t) curve becomes sigmoidal for larger t.
But also in this case deracemization proceeds faster. In addition,
we showed that complete deracemization can be achieved by
Viedma ripening without the need of cluster incorporation
effects, when chiral additives induce a difference in growth rate
between both enantiomers.
APPENDIX C: APPROXIMATE EXPRESSION FOR
THE DERACEMIZATION RATE CONSTANT
■
The set of coupled differential eqs 1a−1d in the main text is
given below once more, but now with c = 0 and instantaneous
racemization in solution. This set cannot be solved analytically
in an exact manner. To find an expression for the deracemiza-
tion rate constant k, approximations have to be made.
∂B+
= a+B+(M − MeBq) − bB+
(C1a)
∂t
∂C+
∂t
= bB+ + a+C+(M − MeCq)
(C1b)
with analogous equations for the opposite enantiomer.
∂M
= −a+B+(M − MeBq) − a+C+(M − MeCq)
∂t
− a−B−(M − MeBq) − a−C−(M − MeCq)
(C1c)
APPENDIX A: VIEDMA RIPENING EXPERIMENTS
WITH ADDITIVES
■
T = B+ + C+ + B− + C− + M
(C1d)
In acetonitrile (ACN), the linear increase in ee was reproducibly
found for different amounts of additive (Figure 9).
After premixing as described in the main text, at the start
+
−
(t = 0) of the deracemization simulation run B0 = B0 = B0,
C 0+ = C 0− = C 0 and ee(t = 0) = 0. Careful examination of the
We found that deracemization of compound 1 in methanol
(MeOH) instead of ACN again proceeded in a linear fashion
when additives were present (Figure 10). The racemization rate
of 1 was found to be significantly slower in MeOH, which is
probably due to hydrogen bonding interactions with the solvent
(Figure 11, Appendix B). The half-life time of racemization in
ACN (68 min) was found to be 6 times smaller than in
evolution of B+, B−, C+ and C reveals a linear behavior for
−
t not too large for nearly all conditions (Figure 12) according to
B+ = B0(1 + pt)
(C2a)
B− = B0(1 − pt)
(C2b)
F
DOI: 10.1021/acs.cgd.5b00127
Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design
Article
Figure 10. Deracemization experiments conducted in MeOH. (a) Linear increase in ee for different amounts of additive. (b) Rate of deracemiza-
tion as a function of the amount of additive calculated from linear fits of the data in panel a. The scales are different as compared to
Figures 2b and 9.
+
−
As (B+ + B− + C + C ) = 2(B0 + C ) is constant,
0
+
1
∂B
∂C+
∂t
∂B−
∂t
∂C−
∂t
⎡
⎢
⎤
∂ee(t)
∂t
=
=
+
−
−
⎥
⎦
⎣
2(B0 + C0) ∂t
(C3)
(C4)
(C5)
or
so
∂ee(t)
∂t
1
[B0 + C0 + B0 + C0]p
2(B0 + C0)
∂ee(t)
∂t
= p = k
So, p is identical to the rate constant k, we searched for.
To obtain an equation for p, we first need an expression for
ΔMB = M − MB,eq. Using the fact that (∂B+/∂t) = −(∂B−/∂t)
and eqs C1a−C1b, we get
Figure 11. Racemization of compound 1 in a stagnant solution of
MeOH (t1/2 = 419.79 min) and ACN (t1/2 = 68.53 min) in the
presence of DBU (16.9 mol %). The data points are fitted with
asymptotic functions from which the racemization half-lives were
determined.
(a+B+ΔMB − bB+) = −(a−B−ΔMB − bB−)
(C6)
or
ΔMB[a+B0(1 + pt) + a−B0(1 − pt)] = 2bB0
(C7)
or
ΔMB[(a+ + a−) + (a+ − a−)pt] = 2b
So, for pt ≪ 1 and (a+ − a−) ≪ (a+ + a−)
(C8)
ΔMB[(a+ + a−)] ≅ 2b
(C9)
this gives
2b
+
+
Figure 12. (a) Evolution of B+, C , B+ + C , and M as a function of
time for the + enantiomer. (b) Evolution of B−, C −, B−+ C −, and M
as a function of time for the − enantiomer. Here a+ > a−.
ΔMB =
(a+ + a−)
(C10)
Now, we are searching for k. Using eq C1a and the fact that
(∂B+/∂t) = pB0 we obtain
C+ = C0(1 + pt)
∂B+
(C2c)
= (a+ΔMB − b)B+ = pB0
(C11)
∂t
C− = C0(1 − pt)
(C2d)
or
We use these semiempirical expressions as a starting point in
deriving an expression for k.
(a+ΔMB − b)B0(pt + 1) = pB0
(C12)
G
DOI: 10.1021/acs.cgd.5b00127
Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design
As pt ≪ 1 we now obtain
a+ΔMB − b = p
Article
(20) Iggland, M.; Mazzotti, M. Cryst. Growth Des. 2011, 11, 4611−
4622.
(21) Skrdla, P. J. Cryst. Growth Des. 2011, 11, 1957−1965.
(C13)
(C14)
́
(22) Iggland, M.; Fernandez-Ronco, M. P.; Senn, R.; Kluge, J.;
and using the relation C10 for ΔMB this gives
2b
Mazzotti, M. Chem. Eng. Sci. 2014, 111, 106−111.
(23) Spix, L.; Alfring, A.; Meekes, H.; van Enckevort, W. J. P.; Vlieg,
E. Cryst. Growth Des. 2014, 14, 1744−1748.
(24) van der Meijden, M. W.; Leeman, M.; Gelens, E.; Noorduin, W.
L.; Meekes, H.; van Enckevort, W. J. P.; Kaptein, B.; Vlieg, E.; Kellogg,
R. M. Org. Process Res. Dev. 2009, 13, 1195−1198.
p = a+
− b
(a+ + a−)
or
(25) Hein, J. E.; Huynh Cao, B.; Viedma, C.; Kellogg, R. M.;
Blackmond, D. G. J. Am. Chem. Soc. 2012, 134, 12629−12636.
(26) Suwannasang, K.; Flood, A. E.; Rougeot, C.; Coquerel, G. Cryst.
Growth Des. 2013, 13, 3498−3504.
(a+ − a−)
p = k = b
(a+ + a−)
(C15)
(27) Suwannasang, K.; Coquerel, G.; Rougeot, C.; Flood, A. E. Chem.
Eng. Technol. 2014, 37, 1329−1339.
AUTHOR INFORMATION
Corresponding Author
*E-mail: w.vanenckevort@science.ru.nl.
■
(28) Addadi, L.; Berkovitchyellin, Z.; Domb, N.; Gati, E.; Lahav, M.;
Leiserowitz, L. Nature 1982, 296, 21−26.
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS
■
Financial support for this work was provided by The Dutch
Astrochemistry Network financed by The Netherlands
Organisation for Scientific Research (NWO).
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H
DOI: 10.1021/acs.cgd.5b00127
Cryst. Growth Des. XXXX, XXX, XXX−XXX