Anomalous size distribution of chiral crystals during deracemization by

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Anomalous size distribution of chiral crystals during deracemization by grinding Hiroyasu Katsuno, and Makio Uwaha Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.9b00095 • Publication Date (Web): 21 Feb 2019 Downloaded from http://pubs.acs.org on February 23, 2019

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Crystal Growth & Design

Anomalous size distribution of chiral crystals during deracemization by grinding Hiroyasu Katsuno∗,† and Makio Uwaha∗,‡ † Department of Physical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan ‡ Science Division, Center for General Education, Aichi Institute of Technology, 1247 Yachigusa, Yakusa-cho, Toyota, Aichi 470-0392, Japan E-mail: [email protected]; [email protected]

Abstract With the use of a generalized Becker-Döring model, we study the anomalous crystal size distribution (CSD) of chiral crystals during deracemization (chirality conversion) under grinding. Ultrasound grinding yields a large amount of small clusters, and large crystals can grow due to the temporal large supersaturation, which leads to the anomalous spread of the CSD. Glass bead grinding produces crystals of various sizes and the steady CSD is maintained. The amplification of the crystal enantiomeric excess is realized in both models. Therefore, temporal spread of the CSD observed in experiment is not an essential condition for the deracemization.

Introduction In 2005, Viedma demonstrated the conversion from a racemic mixture of sodium chlorate crystals to crystals with single chirality by stirring a saturated solution with glass beads. 1

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Chiral crystals are ground by glass beads, and the crystal enantiomeric excess (CEE) is amplified exponentially with time. Noorduin et al. observed a similar behavior for organic chiral molecules with a racemization agent. 2 The deracemization of crystals by grinding has been observed in other chiral crystals. 3,4 The achievement of single chirality by grinding is distinguished from the usual Ostwald ripening and is called Viedma ripening. Several experimental methods of modified Viedma ripening are proposed. Using chiral additives, 5 irradiating polarized light, 6 and using a metastable conglomerate as an initial material 7,8 are successful in acceleration of the CEE amplification. Viedma ripening is also observed under ultrasound grinding instead of glass bead grinding. 9,10 A combination of ultrasound grinding and glass bead grinding leads to faster achievement of single chirality. 11 In the absence of grinding, it was found that temperature cycling also leads to the complete chirality conversion. 12 Viedma ripening is expected as a useful method to realize homochiral crystals with many modifications to be developed in the future. To explain Viedma ripening, various theoretical ideas have been proposed such as cluster incorporation, 13–23 catalytic surface reaction, 24 and mutual inhibition. 25,26 These models have common features, which are autocatalytic and contain a recycling process. 27,28 Although they can reproduce the exponential amplification of the CEE with time as observed in Viedma ripening, it is difficult to identify the underlying mechanism directly in experiment. So far the cluster incorporation model is the only model that explains various features of the deracemization (chirality conversion) both under grinding and temperature cycling. However, there is a discrepancy between theory and experiment about time change of the crystal size distribution (CSD). In the theoretical study based on the rate equation models, 13,14,17,29,30 it is assumed that the CSD does not change under grinding. This assumption is confirmed by numerical calculation of a the Becker-Döring type model with a specific grinding model. 15 In experiment by Hein et al., racemic mixture crystals in a solution is subjected to sonication in the presence of glass beads and their CSD was counted. The obtained CSD near the completion of the CEE amplification is spreaded in comparison with

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the CSD at the initial stage and that after the completion of the CEE amplification. 31 The result is contradictory to the theory although the anomalous CSD may not be always the case. 10 Recently, Xiouras et al. 32 reported the CSD under ultrasound grinding and under glass bead grinding. The ultrasound grinding yields small crystals of 0.1–12 µm and unbroken crystals of several hundreds µm. On the other hand, the glass bead grinding makes a strong size reduction around several hundreds µm and yields small size crystals of the wide range around 1.5–50 µm. The authors characterize the glass bead grinding as fracture and the ultrasound grinding as abrasion. In this paper, we study the time change of the CSD during the CEE amplification based on the chiral cluster incorporation model. 15,33 We focus on the difference of the size distribution of small crystals under two grinding methods, glass bead and ultrasound. According to the experimental observation, 32 we propose models of the ultrasound grinding and the glass bead grinding in our Becker-Döring type model. 33 We will show that the anomalous spread of the CSD, which appears in the ultrasound grinding, represents the occurrence of a temporal high supersaturated state although it is not a necessary condition for the deracemization.

Generalized Becker-Döring model In order to study the time development of the CSD during the CEE amplification, we use the Becker-Döring type model. In the original Becker-Döring model, monomers are special because the CSD can change only through incorporation and emission of monomers. 34–36 In this study, incorporation and emission of clusters, which are very small, are considered. Thus the system has many growth units: monomers and small clusters. 13 The basic idea of the extension of the Becker-Döring model is the same as our previous work for grinding 15 and for temperature cycling. 33 We introduce chiral types, right and left, denoted by α=R or L, and cluster incorporation up to a certain size is , which is the number of molecules. The time development of the number of clusters of the size i is written as (see Supporting

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Informaion) ∂nαi = ∂t

(

∂nαi ∂t

)

( +

a

∂nαi ∂t

)

( +

r

∂nαi ∂t

) ,

(1)

g

where the subscript "a" represents attachment/detachment, "r" racemization, and "g" grinding. Hereafter, we call clusters of the size i = 1 monomers, those of the size 2 ≤ i ≤ is small clusters, and those of the size is < i large crystals. The time change of nαi by attachment is proportional to the numbers of the colliding clusters, σi,j nαi nαj (σi,j represents the collision rate). The rate of detachment λi,j is determined from the detailed balance condition: α,eq σi,j nα,eq nα,eq = λi+j,j nα,eq represents the number at equilibrium). Racemization is i j i+j (ni

assumed to occur in the monomer level at the rate r, and the net racemization is determined from the difference of the number of R type monomers and that of L type monomers L r(nR 1 − n1 ).

Figure 1: Schematic figure of our grinding models of ultrasound and glass beads. In this study, we consider two types of grinding (Figure 1). According to the work by Xiouras et al., 32 the ultrasound grinding yields many tiny fragments. When the crystal size becomes larger than a threshold size ig , the crystal can be scraped at the rate g, and small fragments appear. Small fragments are distributed uniformly in size with the mean size if 4


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Table 1: All reactions in our model. For grinding, the parameter values of if and ∆f used are shown. λi+j,j

↼ −− − − Ri+j Ri +Rj − − ⇁

attachment/detachment of size j ≤ is

σi,j λi+j,j

↼ −− − − Li+j Li +Lj − − ⇁ σi,j r

↼ − L1 R1 ⇁ −

racemization

1 Ri ⇒ = 17 [(R2 + · · · +R18 ) + (Ri−18 + · · · +Ri−2 )] g 1 Li ⇒ = 17 [(L2 + · · · +L18 ) + (Li−18 + · · · +Li−2 )]

grinding by ultrasound (ig ≤ i ≤ imax )

r

g

g

Ri ⇒ = g = Li ⇒

2 (R2 i−3 2 (L2 i−3

+ · · · +Ri−2 ) + · · · +Li−2 )

grinding by glass beads (ig ≤ i ≤ imax )

and the dispersion ∆f . Simultaneously, larger fragments also increase as shown in Figure 1. The glass bead grinding provides crystals of various sizes from a larger one (Figure 1). From a crystal larger than the threshold size ig , crystals of the size 2 ≤ i ≤ ig − 2 are produced by glass bead grinding. The concrete form of the grinding model is summarized in Supporting Information. All reactions in our model are listed in Table 1 using the R and L type crystal of size i, Ri and Li . Our choice of parameters in the simulation is as follows: the number of monomers at equilibrium is nR,eq = nL,eq = 10−3 , the effective surface tension α ¯ = 10, the maximum 1 1 crystal size imax = 200 and the maximum size of small clusters is = 20. The total mass is set to unity. The grinding rate is g = 0.05, the threshold grinding size is ig = 100 and the mean size of small fragments and its dispersion are if = 10 and ∆f = 8, i.e., the size from i = 2 to i = 18 are equally produced in ultrasound grinding.

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Results & Discussion We first look at the time change of the CEE, which is defined as the ratio of the difference of the masses of large crystals of R type and L type to the total crystal mass: imax ∑

ϕ=

L i(nR i − ni )

i=is +1 imax ∑

(2)

. i(nR i

+

nLi )

i=is +1

The initial distribution is uniform in size and the initial CEE is set to 0.1 in this study. Figure 2 shows the time change of the CEE, which increases exponentially under both grinding methods. The complete chirality conversion is accomplished around t ≃ 8 × 104 for ultrasound grinding and t ≃ 8 × 105 for glass bead grinding. The CEE amplification rate of the ultrasound grinding is 10 times larger than that of the glass bead grinding in our numerical simulation. In an experiment, 11 the ultrasound grinding alone cannot realize the homochiral state since the ultrasound cannot break initial large crystals effectively. 32 Large amplification rate by ultrasound grinding in our simulation may be resulted from the smallness of the maximum crystal size (imax = 200) in our numerical study. 1

0.8

0.8

0.6

0.6

φ

1

φ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4

0.4

0.2

0.2

0

(a)

0

2 × 104 4 × 104 6 × 104 8 × 104 1 × 105

t

0

(b)

0

2 × 105 4 × 105 6 × 105 8 × 105 1 × 106

t

Figure 2: Time change of the CEE for (a) ultrasound grinding and (b) glass bead grinding.

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Ultrasound grinding We first investigate the CSD under ultrasound grinding. The initial artificial CSD is relaxed to a natural state within t ∼ 104 with our parameter choice. The CSD after the initial relaxation at t = 2 × 104 (ϕ ∼ 0.2), the CSD near the end of deracemization at t = 6 × 104 (ϕ ∼ 0.8), and the CSD of the steady state at t = 1.5 × 104 (ϕ = 1) are shown in Figure 3. Red area and green area represent the number of R type and L type crystals. Blue circles represent the total number of the crystal of size i. At t = 2 × 104 [Figure 3(a)], the distributions of both types have the maximum peaks at sizes slightly smaller than the threshold grinding size ig = 100. Although the CEE is about ϕ = 0.2, the number of L type crystals is larger than that of R type for 50 ≤ i ≤ 70. As time passes, the number of R type crystals increases and that of L type decreases. At t = 6 × 104 [Figure 3(b)], the CEE is about 0.8, and the small number of large L type crystals remain around i = 90, so that the total crystal size distribution mostly consists of R type crystals. Finally, at t = 1.5 × 105 [Figure 3(c)], large L type crystals disappear and only large R type crystals survive. Figure 4(a) shows the total size distribution of monomers and small clusters 1 ≤ i ≤ 20. Although the number of monomers is almost constant, the total number of small clusters is largest at t = 6 × 104 , and smallest at t = 1.5 × 105 . Figure 4(b) shows the total CSD of large crystals around the maximum peak 80 ≤ i ≤ 150. The distribution of large crystals at t = 2 × 104 (red plus) is almost the same as that at t = 1.5 × 105 (blue cross). At the intermediate time t = 6 × 104 (green circle), the number of crystals around the peak i ∼ 90 decreases and the crystals larger than the threshold size i > ig increases. In this period, the large crystals are growing faster than being ground. Faster growth of crystals indicates that effective supersaturation 37 of the system becomes larger temporarily around t = 6 × 104 , where the CEE is close to unity. To find the key mechanism of the change of the CSD, we focus on the mass of small clusters (2 ≤ i ≤ is = 20) of R type and that of crystals larger than the grinding threshold size (100 = ig ≤ i ≤ imax = 200) of R type. Figure 5(a) shows the time change of the mass 7


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7

6

6

4

4

3

3

2

1

1

0

0

(a)

4

3

2

0

50

100

150

i

200

5

Number ×10

5

Number ×10

5

4

7

6

4

7

Number ×10

2 1

(b)

0

0

50

100

150

i

200

(c)

0

50

100

i

150

200

Figure 3: The CSD at (a) t = 2 × 104 , (b) t = 6 × 104 , and (c) t = 1.5 × 105 under ultrasound grinding. Red (black) and green (gray) areas represent the number of crystals of R type and that of L type. Blue (black) circles represent the total number of crystals.

10-2

7

t = 2 ×1044 t = 6 ×10 t = 1.5 ×105

10-4

4

10

10-5 -6

10

-7

10

5 4 3 2 1

-8

0

10

(a)

t = 2 ×1044 t = 6 ×10 t = 1.5 ×105

6

Number ×10

-3

Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4


2

4

6

8

10

i

12

14

16

18

20

(b)

80

90

100

110

i

120

130

140

150

Figure 4: Total CSD at t = 2 × 104 (red plus), t = 6 × 104 (green circle), and t = 1.5 × 105 (blue cross). The crystal size range is (a) 1 ≤ i ≤ 20 and (b) 80 ≤ i ≤ 150.

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of small clusters of R type (102 times enlarged) and that of large crystals, and Figure 5(b) shows the time change of mass flow from R type monomers to L type monomers, that is, the rate of the molecular chirality conversion (deracemization). During the CEE amplification

Anomalous size distribution of chiral crystals during deracemization by

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