A Cooperative Kinetic Model to Describe Crystallization in Solid

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A Cooperative Kinetic Model to Describe Crystallization in Solid Solution Forming Systems Maksymilian Olbrycht, Maciej Balawejder, Izabela Poplewska, Heike Lorenz, Andreas Seidel-Morgenstern, Wojciech Piatkowski, and Dorota Antos Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b01768 • Publication Date (Web): 10 Jan 2019 Downloaded from http://pubs.acs.org on January 14, 2019

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Crystal Growth & Design

A Cooperative Kinetic Model to Describe Crystallization in Solid Solution Forming Systems Maksymilian Olbrychta, Maciej Balawejderb, Izabela Poplewskaa, Heike Lorenzc, Andreas Seidel-Morgensternc,d, Wojciech Piątkowskia, Dorota Antosa,* a Department

of Chemical and Process Engineering, Faculty of Chemistry, Rzeszow University of Technology,

35-959 Rzeszow/PL b Chair c Max

of Chemistry and Food Toxicology, University of Rzeszow, 35-959 Rzeszow/PL

Planck Institute for Dynamics of Complex Technical Systems, 39106 Magdeburg/DE

d Institute

of Process Engineering, Faculty of Process and Systems Engineering, Otto von Guericke University

Magdeburg, 39106 Magdeburg/DE

Abstract The crystallization kinetics was determined for mixtures of stereoisomers which revealed solid solution phase behavior in the crystalline phase. Two model systems of pharmaceutically active compounds were considered: stereoisomeric salts of citalopram and nafronyl. The kinetic data were acquired for seeded as well as unseeded crystallization. In the former case, a small amount of seed crystal was added into liquid solutions to alter the crystallization progress. Both systems differed markedly in the course of solid-liquid equilibrium data and crystallization kinetics. To quantify the progress of the process, a model for crystallization kinetics was developed along with a procedure for determination of underlying kinetic parameters. The moment method was exploited, which was modified to account for nucleation and growth of mixed crystals. The model was capable of predicting crystallization kinetics in both systems despite the differences between them. Therefore, it can potentially be adopted to describe crystallization kinetics for other solid solution forming systems.

*

Corresponding authors:

[email protected], phone.: +48 178651853; fax: +48 178543655

1 ACS Paragon Plus Environment

Crystal Growth & Design 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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1. Introduction A significant number of the drugs currently in use contain chiral substances as active pharmaceutical ingredients (API).1,2 Most isomers of chiral API exhibit marked differences in biological activities which determine pharmacological actions of the drug. Therefore, a considerable interest has arisen in pharmaceutical companies in chiral separations in order to eliminate the unwanted isomer from the preparation.2 The most frequently used method for chiral separations is the so-called classical resolution method, which involves formation of diastereoisomeric salts by reaction with a single-enantiomer resolving agent. The diastereoisomers are then separated by conventional crystallization, taking advantage of the differences in their solubility. The efficiency of the operation is determined by the solid phase behavior of chiral systems.3 The method is the most effective and straightforward for the design when isomers form a mixture of individual crystals, i.e., they are completely immiscible in the solid phase.4 In such a case, a single-stage crystallization can provide pure crystals of the target compound. The separation is much more challenging when solid solutions are formed, which occurs for components with miscibility in the solid phase. Though mixtures of chiral organic compounds rarely exhibit continuous solid solutions, partial solid solutions are reported to occur relatively often.5-17 The separation of such mixtures requires multistage crystallization, in which the target compound can be enriched either in the solid or in the liquid phase, depending on the solubility properties of the mixture. Multistage crystallization in solid solution forming systems has been described in several studies,18-25 in which the process design was based on the solid-liquid equilibrium (SLE) data. Up to now, the issue of crystallization kinetics in such systems has not been tackled quantitatively. Nevertheless, determination of crystallization kinetics is a prerequisite for successful application of the design procedure to industrial-scale separations. This is particularly important when a long period of time is needed to establish crystallization equilibrium. Then, the kinetic analysis is indispensable for the optimization of the efficiency of multistage operation. Therefore, in this work we performed a study on crystallization kinetics for solid solution forming systems. Two different model systems were analyzed, which consisted of pharmaceutically active stereoisomers that showed miscibility in the solid phase, i.e., System I, which was a mixture of diastereoisomeric salts of citalopram with (+)-O,O′-di-p-


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toluoyl-D-tartaric acid, ((+)DTT)), and System II, which was a mixture of stereoisomeric salts of nafronyl with oxalic acid. Citalopram,

1-[3-(dimethylamino)-propyl]-1-(p-fluorophenyl)-5-phthalancarbonitrile

(Fig. 1A), is an antidepressant drug used to treat depression. The S-enantiomer (escitalopram) was proved to possess much higher biological activity compared to the Renantiomer of that compound.26 (+)DTT salts of citalopram exhibited the solid solution phase behavior almost over entire composition range.22,27 In a previous study, we used a multistage batch-wise crystallization process for resolving binary mixtures of the citalopram diastereoisomers to obtain pure S-citalopram. 22 Nafronyl

oxalate,

oxalate

salt

of

2-(diethylamino)ethyl

3-(naphthalen-1-yl)-2-

((tetrahydrofuran-2-yl)methyl)propanoate (Fig. 1B), is a pharmacologically active compound, which is also used as a drug ingredient, e.g., for the treatment of vascular diseases.28-30 The nafronyl molecule possesses two stereogenic centers. It is manufactured as a stereoisomeric mixture of two pairs of racemates being diastereoisomers to each other. The mixture forms solid solutions in the crystalline phase. Multistage batch-wise crystallization was exploited to resolve pseudo-binary mixtures of the two racemates, one of which was the target product that contained the most active stereoisomer.24,25

F

F

N

N

O N

(R)-( )

O N

(S)-(+)

Fig. 1A. Chemical structure of R- and S-citalopram.



C2

C2'

O

N

*

*

O

N O

C2

C2'

*

*

N O O 2R; 2'S (2)

O 2R; 2'R (1)

C2

C2'

*

*

O O

C2

C2'

*

*

O

2S; 2'S (3)

Page 4 of 26

O

N O

Racemate 1,3

O 2S; 2'R (4)

Racemate 2,4

Fig. 1B. Chemical structure of nafronyl along with the nomenclature used for the stereoisomers.

Though stereoisomeric mixtures of both compounds form continuous solid solutions in the crystalline phase, they differ markedly in the crystallization mechanism. The crystallization of citalopram diastereoisomers occurred in a relatively typical manner, in which reproducible equilibrium states were established regardless of the composition of the mixture.22 Nafronyl oxalate revealed a complex phase behavior, which manifested itself in multiple equilibrium states, which established over a certain composition range of the mixtures.25 The occurrence of a desired equilibrium state could be imposed by seeding the supersaturated solutions. In both cases we put efforts in order to: a) formulate a kinetic model capable of reproducing the phase behavior, including the effect of seeding on the crystallization course; b) provide a procedure for the determination of underlying model parameters; c) validate the model. The model accounted for “cooperative” crystallization in solid solutions, in which contribution of all mixture components to nucleation and crystal growth was accounted for. The kinetic parameters were determined by matching the model solutions and selected sets of kinetic data. Next, the model was successfully verified by comparing the prediction results to corresponding experimental data over a wide range of the mixture compositions. The general goal of the study was to fill the gap in the existing literature, which lacks mathematical description of crystallization kinetics in solid solution forming systems. The specific goal was to provide an efficient mathematical tool that can potentially be used to


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optimize the efficiency of crystallization for resolving stereoisomeric mixtures of citalopram and nafronyl, which both are industrially relevant.

2. Moment model The progress of crystallization process is usually quantified using a population balance model that accounts for nucleation and crystal growth kinetics. The population balance equations can be formulated in a convenient way based on the moment analysis, as follows:3 𝑑 𝑛 (𝑡) 𝑑𝑡

= 𝑛 𝐺0 (𝑡) 𝑛−1 (𝑡) + (𝐿𝑐𝑟𝑖𝑡 )𝑛 𝐵0 (𝑡)

(1)

where: t is the time coordinate; B0 is the nucleation rate; G0 is the overall crystal growth rate; Lcrit is the critical diameter of the nucleus at the initial supersaturation; n is the order of the moment. Each n-th moment implies from the previous (n −1)-th moment, apart from the zeroth moment (n = 0), for which the first term on the left-hand side of Eq. (1) is canceled, while the second one is active, since (𝐿𝑐𝑟𝑖𝑡 )0 = 1. The value of Lcrit is very small, therefore for all moments n > 0 the second term on the left-hand side of Eq. (1) can be neglected. The zeroth moment, 0, represents the overall number of crystals; 1 corresponds to the total length of the crystals; 2 – to the total surface of the crystalline phase; 3 – to the total volume of the crystalline phase. Eq. (1) is valid for a mixture of compounds that are immiscible in the solid phase, and appear in the form of individual crystals. Crystals of miscible compounds are of multicomponent nature. To account for that phenomenon, two moment equations were formulated, which describe the total moments of the crystalline phase (Eq. (2)), and the individual moments for each component i (Eq.(3)): 𝑑 𝑛,𝑝 (𝑡) 𝑑𝑡

𝑁𝐶 𝑛 = 𝑛 (𝑛−1),𝑝 (𝑡) ∑𝑁𝐶 𝑖=1 𝐺0,𝑝,𝑖 (𝑡) + (𝐿𝑐𝑟𝑖𝑡 ) ∑𝑖=1 𝐵0,𝑝,𝑖 (𝑡)

𝑑 𝑛,𝑝,𝑖 (𝑡) 𝑑𝑡

= 𝑛 (𝑛−1),𝑝 (𝑡)𝐺0,𝑝,𝑖 (𝑡) + (𝐿𝑐𝑟𝑖𝑡 )𝑛 𝐵0,𝑝,𝑖 (𝑡)

(2) (3)

The set of Eqs (2) and (3) applies for multicomponent solid solutions consisting of the NC number species i, which can form the NP number of solid state forms (e.g. polymorphs), p. Accordingly, n,p denotes the total n-th moment of the crystalline phase; n,p,i is the individual n-th moment of the component i. In the specific cases under this study it holds:



-

Page 6 of 26

for System I that consists of binary mixtures, NC = 2, without the presence of polymorphs, NP = 1,

-

for System II that consists of pseudo-binary mixtures, NC = 2, the presence of two polymorphs is accounted for, NP = 2.

The moment model represented by Eqs (2-3) is formulated assuming “cooperative” contributions of kinetic rates of all components of the crystalline phase to the values of the total moments. This means that the mixed crystals can be formed from nuclei of each component, and the growth of their length, surface and volume results from the contribution of growth kinetics of each component. The third moment can be used to calculate the mass of each component in the solid phase, 𝑚𝑖𝑆 formed during the course of crystallization: 𝑑𝑚𝑖𝑆 (𝑡) 𝑑𝑡

= 3 𝑘𝑣 𝑆 2,𝑝 (𝑡)𝐺0,𝑖,𝑝 (𝑡)

(4)

where kv is the volume shape factor; S is the solid phase density. The mass fraction of the component i in the mother liquor, 𝑥𝑖𝐿 , and in the solid phase, 𝑥𝑖𝑆 , can straightforwardly be determined as follows: 𝑥𝑖𝐿 (𝑡) = 𝑥𝑖𝑆 (𝑡) =

𝑚𝑖𝐹 − 𝑚𝑖𝑆 (𝑡)

(5)

𝑚𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑚𝑖𝑆 (𝑡)

(6)

𝑆 ∑𝑁𝐶 1 𝑚𝑖 (𝑡)

where 𝑚𝑖𝐹 is the mass of the component i in the feed. The model has to be supplemented with kinetic rate equations of nucleation and crystal growth (G0,p,i and B0,p,i), which are specific for each mixture component and polymorph. In this study, a kinetic equation of secondary nucleation was adopted:3 𝐿 𝐵0,𝑝,𝑖 = 𝑘𝐵,𝑝,𝑖 (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 )

𝑟𝑝,𝑖

(7)

where kB,p,i is the nucleation rate constant; r p,i is the empirical exponent (equation order); 𝐿 both values are characteristic for the component i and the polymorph p; 𝑥𝑒𝑞,𝑖 , corresponds

to the mass fraction of the component i in the mother liquor at crystallization equilibrium. The crystal growth kinetics was described by the following kinetic equation: 𝐿 𝐺0,𝑖,𝑝 = 𝑘𝐺,𝑝,𝑖 (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 )

𝑞𝑝,𝑖

(8)

where kG,p,i is the rate constant of crystal growth; qp,i is the empirical exponent (equation order). The rate constants, kB and kG, are a function of the hydrodynamic conditions and the process temperature, which were kept invariant in this study. 6 ACS Paragon Plus Environment

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In both kinetic equations Eqs (7) and (8), the driving force is the difference between the temporary concentration of the component i in the mother liquor, 𝑥𝑖𝐿 , and the equilibrium 𝐿 concentration, 𝑥𝑒𝑞,𝑖 . The latter is always a function of the mass of the solvent added, even

for solutions having the same diastereoisomeric excess. Therefore, for solid solution forming systems the meaning of the equilibrium concentration diverges from typical definition of solubility. The equilibrium concentration is determined by the following set of the mass balances (9-12) coupled with the closure equation, Eq. (13), and the SLE relationship, Eq.(14) (Fig. 1):22-25 𝑚𝐹 + 𝑚 𝑆𝑜𝑙 = 𝑚𝑀

(9)

𝑚 𝑆 + 𝑚𝐿 = 𝑚𝑀

(10)

𝑚𝐹 𝑥𝑖𝐹 = 𝑚𝑀 𝑥𝑖𝑀

(11)

𝐿 𝑚 𝑆 𝑥𝑖𝑆 + 𝑚𝐿 𝑥𝑒𝑞,𝑖 = 𝑚𝑀 𝑥𝑖𝑀

(12)

𝑆 ∑𝑁𝐶 𝑖 𝑥𝑖 = 1

(13)

𝐿 𝑥𝑒𝑞,𝑖 = 𝑓𝑝 (𝑥𝑖𝑆 )

(14)

where F is the mass of feed delivered to the crystallizer; mL – the mass of the mother liquor; mS – the mass of the crystalline phase; mSol – the mass of the solvent; mM – the 𝐿 whole mass of the system obtained after mixing (at the mixing point); the function 𝑥𝑒𝑞,𝑖 =

𝑓𝑝 (𝑥𝑖𝑆 ) is specific for each of components, i, and each of polymorphs, p (in case when they are formed, i.e. for System II). The amount of feed mF, and its composition 𝑥𝑖𝐹 , are known a priori. The solution of the set of Eqs (9-14) provides the values of all unknown variables along with the equilibrium 𝐿 compositions in both phases, 𝑥𝑒𝑞,𝑖 , 𝑥𝑖𝑆 . The streams incoming and outcoming from the

crystallizer and their compositions are illustrated in an explanatory diagrams depicted in Figs 2A and 2B. The model equations, Eqs (1-14), have to be supplemented with initial conditions, adequate to the crystallization mode, i.e., seeded or unseeded crystallization. In case of seeded crystallization, all moments of seeds added into the feed solution have to be specified regarding their total length, surface and volume. For the unseeded crystallization all moments are initially equal to zero.



A)

B)

Page 8 of 26

Sol

solubility line, conodes,

Sol F,

mixing line. S,

M

L,

D1

D2

Fig. 2. Illustration of the streams incoming and outcoming from a crystallization unit on a flowsheet scheme A), and on the ternary solubility phase diagram B).

3. Experimental 3.1. Substances As mentioned above, two different model systems were used for the study of crystallization kinetics in solid solution forming systems: -

System I: binary mixtures of diastereoisomeric salts of R,S-citalopram·(+)DTT (Fig. 1A), where the target compound was S-citalopram·(+)DTT, which was enriched in mother liquor;

-

System II: quaternary mixtures of stereoisomers of nafronyl oxalate (Fig. 1B), where the target was the racemate (2R,2′S)/(2S,2′R) containing the most active stereisomer (2S,2′R), which was enriched in the crystalline phase.

Both systems were chemically different and required different solvent environments for the realization of experiments. The chemicals used for the crystallization and the product analysis were as follows. System I: racemic citalopram hydrobromide, i.e., (R,S)-1-[3(dimethylamino)propyl]-1-(4fluorophenyl)-1,3-dihydroisobenzofuran-5

carbonitrile

hydrobromide

(molecular

formula C20H22BrFN2O) with minimum purity 99%, purchased from Jubilant Organosys Ltd.; escitalopram oxalate, i.e., (S)-1-[3-(dimethylamino)propyl]-1-(4-fluorophenyl)-1,3dihydroisobenzofuran-5-carbonitrile oxalate (molecular formula C22H23FN2O5) with purity Pu ≥ 98%, purchased from Yick-Vic Chemical & Pharmaceuticals Ltd.; (+)DTT, i.e., 8 ACS Paragon Plus Environment

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(+)-O,O′-di-p-toluoyl-D-tartaric acid (molecular formula C20H18O8), with minimum purity 98% delivered by Sigma-Aldrich. To perform crystallization experiments, acetonitrile with HPLC-grade (VWR Prolabo), and methanol with HPLC-grade (Fisher Chemical) were used. For the HPLC analysis methanol with HPLC-grade (Fisher Chemical), triethylamine with purity Pu ≥ 99.5% (Sigma-Aldrich), and anhydrous acetic acid were used. System II: nafronyl oxalate, i.e., the oxalate salt of 2-(diethylamino)ethyl 3-(naphthalen-1yl)-2-((tetrahydrofuran-2yl)methyl)propanoate

(molecular

formula

C26H35NO7,)

purchased from Santa-Cruz Biotechnology (USA) in the form of a stereoisomeric mixture with HPLC purity of 98%. To perform crystallization experiments, acetone with HPLCgrade (POCH, Poland) was used. For the HPLC analysis n-hexane, propan-2-ol with HPLCgrade (POCH), and diethylamine with minimum purity 99.5% (Sigma-Aldrich), were used.

3.2. Procedures 3.2.1. HPLC analysis System I The HPLC analysis of citalopram solutions was performed using a Chirobiotic V column (Astec Co.) with dimensions 250 × 4.6 mm, and the adsorbent particle size of 5 μm. The chiral selector used in the column was macrocyclic antibiotic vancomycin, which was immobilized onto a silica matrix. The mobile phase composition was a mixture of

methanol,

trimethylamine,

anhydrous

acetic

acid

with

the

composition:

99.9/0.06/0.055 v/v. The measurements were performed at a flow rate of 1 [mL min-1] and temperature of 25 °C, the injection volume was 5 μL. The UV signal was recorded at a wavelength of 240 nm. All samples for the HPLC analysis were prepared by their dissolution in the eluent. The samples of the crystalline phase were prepared by dissolving up to a concentration of 1 [g L-1], whereas mother liquors were acquired with the amount of 500 μL and diluted 10 – 100 times. The elution order was identified in our previous work,22 as follows: S-citalopram·(+)DTT, R-citalopram·(+)DTT. The HPLC analysis for each sample was conducted in triplicate. The maximum value of the standard deviation for the value of diastereoisomeric excess, deS, of the target enantiomer S-citalopram·(+)DTT, did not exceed 3%. The diastereoisomeric excess, deS, of the target diastereoisomer S, was defined as follows:



𝑗

𝑑𝑒𝑆 =

𝑗

𝑗

𝑗

𝑗

(𝑥𝑆 − 𝑥𝑅 ) (𝑥𝑆 +𝑥𝑅 )

100%

j = L, S

Page 10 of 26

(15)

where S and R denote S-citalopram·(+)DTT, R-citalopram·(+)DTT, respectively; j is the phase index: L – the mother liquor, S – the solid phase. System II The HPLC analysis of nafronyl oxalate solutions was performed using a CHIRALPAK IC column (Chiral Technologies Europe, Illkirch Cedex, France) with dimensions 250 × 4.6 mm and the adsorbent particle size of 5 μm. The chiral selector was cellulose tris(3,5-dichlorophenylcarbamate), which was immobilized onto a silica matrix. The mobile phase consisted of n-hexane, propan-2-ol, diethylamine with the composition 90/10/0.1 v/v. The measurements were performed at a flow rate of 1 [mL min-1] and temperature of 25 °C, the injection volume was 10 μL. The UV signal was recorded at a wavelength of 226 nm. All samples destined for the HPLC measurements were dissolved in propan-2-ol. The concentration of samples taken from the crystalline phases was ca. 0.25 [g L-1]. The samples of mother liquors were acquired with the amount of 200 μL and diluted 10 – 100 times. The elution order was identified in our previous work, as follows:24 2R,2'R; 2R,2'S; 2S,2'S; 2S,2'R (Fig. 2B). To simplify the nomenclature, the numbers are assigned to each of stereoisomers, accordingly to their elution order: (2R,2'R) ≡ 1, (2R,2'S) ≡ 2, (2S,2'S) ≡ 3, (2S,2'R) ≡ 4 . The pairs of racemates are termed as: 1,3, and 2,4 (target), respectively (Fig. 1B). The HPLC analysis of System II was conducted with the same accuracy as described for System I. The diastereoisomeric excess, de2,4, of the target racemate 2,4 was defined as follows: 𝑗

𝑑𝑒2,4=

𝑗

𝑗

𝑗

𝑗

(𝑥2,4 − 𝑥1,3 ) (𝑥2,4 + 𝑥1,3 )

100%

j = L, S

(16)

3.2.2. Synthesis routes of citalopram salts Synthesis of R,S-citalopram·(+)DTT The conversion of racemic citalopram hydrobromide into R,S-citalopram·(+)DTT salt was realized in two steps, including: conversion of racemic citalopram hydrobromide into racemic citalopram free base, and subsequent conversion of racemic citalopram free base


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into R,S-citalopram·(+)DTT salt. The details of the synthesis were described in a previous work.22 Synthesis of S-citalopram·(+)DTT The conversion of escitalopram oxalate into S-citalopram·(+)DTT was also performed in two steps: conversion of escitalopram oxalate into S-citalopram free base, and next conversion of S-citalopram free base into S-citalopram·(+)-DTT. The conversion procedures were previously described.22

3.2.3. Pre-separation of the raw material of nafronyl oxalate Nafronyl oxalate is commercially available in the form of a quaternary mixture with 𝑆 𝑆 diastereoisomeric excess 𝑑𝑒2,4 = 44% (𝑥2,4 = 0.72). Therefore, to determine the

crystallization kinetics over a wide range of diastereoisomeric excess, the raw mixture was subjected to preliminary resolution by multistage crystallization. To enrich the mixtures with the racemate 2,4, the crystalline phase was recrystallized from acetone several times, whereas the diastereoisomeric excess of the racemate 1,3 was achieved by multistage crystallization from the mother liquors. The procedures for processing the crystalline phase and mother liquors were developed, and described in detail in previous works.,24,25

3.2.4. Measurements of crystallization kinetics The kinetic profiles were determined in the liquid and solid phases for System I, where the concentration of R and S-citalopram·(+)DTT was monitored, and for System II, in which the concentration of the racemates 2,4 and 1,3 was measured. The unseeded and seeded crystallizations were performed, to determine the effect of seeding on the crystallization course in both Systems I and II. The measurements procedures are specified for each of systems as follows. System I Three stock mixtures of the crystalline phase were prepared by mixing R,S-citalopram·(+)DTT salt with diastereoisomeric excess 𝑑𝑒𝑆𝑆 = 0% (𝑥𝑆𝑆 = 0.5) with Scitalopram·(+)DTT salt with 𝑑𝑒𝑆𝑆 = 97% (𝑥𝑆𝑆 = 0.985), to obtain the crystalline solutions with diastereoisomeric excesses: 𝑑𝑒𝑆𝑆 = 10% (𝑥𝑆𝑆 = 0.55), 50% (𝑥𝑆𝑆 = 0.75), and 80% (𝑥𝑆𝑆 = 0.90). Each stock mixture was divided into eight batches, 0.2 g each, which were suspended in the same volume of the solvent, i.e., a mixture of acetonitrile and methanol (1/0.08 v/v) in closed glass vials. The volume of the solvent was adjusted according to the 11 ACS Paragon Plus Environment


Page 12 of 26

SLE data previously measured22 to obtain supersaturated solutions. All vials were heated up to 50 °C in water bath for complete dissolution of the solid phase. Next, to initiate crystallization, the vials were placed into a thermostated vessel, and stirred at 25°C with the frequency of rotation 800 rpm. The solutions with 𝑑𝑒𝑆𝑆 = 10% (i.e., prepared from the solid phase with 𝑥𝑆𝑆 = 0.55) and 80% (𝑥𝑆𝑆 = 0.90) were unseeded prior to the crystallization experiments, whereas the solutions with 𝑑𝑒𝑆𝑆 = 50% (𝑥𝑆𝑆 = 0.75) were processed in two series, in which they were either unseeded or seeded using seeds with 𝑑𝑒𝑆𝑆 = 4% (𝑥𝑆𝑆 = 0.52). In the latter case the amount of 1 mg of seed crystals was added to the thermostated vials at the start of the process. Each of eight batches of the same stock was withdrawn from the thermostat in different time intervals, starting after about 4th h, when first crystals appeared, until 72nd h, when the equilibrium was established, i.e., the concentration of stereoisomers in mother liquor in subsequent measurements remained time-invariant. The solutions withdrawn were centrifuged at 25°C, at 4000 rpm for 4 min. Samples of the mother liquor were acquired using a syringe with a filter having anhydrous pores 0.2 µm, and subjected to the HPLC analysis (section 3.2.1). The solid phase was separated out of the mother liquor and dried in a vacuum dryer at temperature 50°C and pressure 0.4 bar, for 24 h. Samples of dried crystalline phase were analyzed by HPLC. Additionally, the density of the solid phase was determined picnometrically; the value obtained was S = 1200 [kg m-3]. System II To measure crystallization kinetics of nafronyl oxalate isomers, four stock solutions were 𝑆 𝑆 𝑆 𝑆 prepared with 𝑑𝑒2,4 = 90% (𝑥2,4 = 0.95), 50% (𝑥2,4 = 0.75), and −20% (𝑥2,4 = 0.40) by 𝑆 𝑆 mixing the raw material with 𝑑𝑒2,4 = 44% (𝑥2,4 = 0.72) with the purified racemate 2,4 with 𝑆 𝑆 𝑆 𝑑𝑒2,4 = 98% (𝑥2,4 = 0.99), or with the material enriched with the racemate 1,3, 𝑑𝑒2,4 = 𝑆 −50%, (𝑥2,4 = 0.25), to obtain the predefined de2,4 ratios. Each stock mixture was divided

into eight batches, 0.2 g each, which were suspended in acetone in closed glass vials. The solvent volume was adjusted according to the SLE data previously measured.24,25 All samples were heated up to 50 °C in water bath for complete dissolution. The batches 𝑆 𝑆 𝑆 𝑆 acquired from the stocks with 𝑑𝑒2,4 = 90% (𝑥2,4 = 0.95) and 𝑑𝑒2,4 = 50% (𝑥2,4 = 0.75) were 𝑆 𝑆 unseeded, while those obtained from the stock with 𝑑𝑒2,4 = −20% (𝑥2,4 = 0.40) were

seeded using the amount of 1 mg of seed crystals with different diastereoisomeric 12 ACS Paragon Plus Environment

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𝑆 𝑆 𝑆 𝑆 excesses: 𝑑𝑒2,4 = 98% (𝑥2,4 = 0.99) or 𝑑𝑒2,4 = −50% (𝑥2,4 = 0.25). Next, the vials were

placed into the thermostated vessel and stirred at 25 °C within different time intervals, i.e., starting after 1st h until 80th h with frequency of rotation 800 rpm. Next, the batches were processed in the same manner, as described above for System I. The density of the solid phase S = 1170 [kg m-3] was determined pycnometrically.

3.2.5. SLE measurements The SLE measurements in both Systems I and II were performed in the previous studies.2225

A number of experimental data were acquired in a wide composition range in the form

of the equilibrium compositions of S- and R-citalopram·(+)DTT or nafronyl racemates 1,3 and 2,4 in the crystalline phase and mother liquors.

3.2.6. XRPD Analysis The XRPD data for both mixtures of Systems I and II were also acquired in the previous studies.24,25 A few measurements were repeated to verify their reproducibility. The measurement procedure is described in the previous works.24,25

4. Results and discussion 4.1. Solid phase behavior The SLE data relevant to the kinetic measurements performed for System I are presented in Fig. 3A, in which the equilibrium liquid phase concentration is plotted against the solid phase concentration of the target product, i.e., S-citalopram·(+)DTT. From Fig. 3A it is evident

that

the

solubility

of

S-citalopram·(+)DTT

is

higher

compared

to

R-citalopram·(+)DTT, therefore, the former can be enriched in the mother liquor. The solubility of S-citalopram·(+)DTT increases continuously with increasing its content in the solid phase, whereas the opposite occurs for R-citalopram·(+)DTT. Both the curves were approximated by polynomial functions (Table 1), which were used to quantify the equilibrium relationship in Eq. (14). The SLE data indicate the formation of solid solutions in the crystalline phase. This was also confirmed by the XRPD measurements;22 the samples of the crystalline phases were characterized by XRPD patterns with similar reflections almost within the whole concentration range, i.e., for 𝑑𝑒𝑆𝑆 < 99% (Fig. 3B). This is characteristic of continuous solid solutions. The equilibrium relationship for the mixture of nafronyl oxalate stereoisomers in System 𝑆 𝐿 II is much more complex (Fig. 4A), particularly in case of the dependency 𝑥𝑒𝑞,1,3 = 𝑓𝑝 (𝑥2,4 ),

for which the equilibrium curve consists of the upward and downward sloping parts. The 13 ACS Paragon Plus Environment


composition region, in which the same liquid concentration can be matched with more than one equilibrium concentration in the solid phase indicates the possibility of establishing multiple equilibrium states. The occurrence of that phenomenon has already been reported in the previous study,25 where we developed a procedure for preseparation stereoisomeric mixtures of nafronyl oxalate by batch-wise multistage crystallization. However, we did not studied there the nature of the phenomenon in detail. This phenomenon can be most probably attributed to polymorphic behavior of one component showing also miscibility at the solid state. This hypothesis is supported by XRPD data, which revealed a small difference in the XRPD pattern between the samples 𝑆 𝑆 with 𝑑𝑒2,4 > 70% and 𝑑𝑒2,4 < 70% (Fig. 4B). Since pure crystals of the 1,3 salt were not

available, this hypothesis cannot be finally clarified yet. However, to quantify the equilibrium relationship for the different solid state forms, the two parts of the SLE curve

A)

0.7

B)

2.5

-4

for the racemate 1,3 were approximated by two different polynomials (Table 1).

2.0

System I 0.6

) (x S =f

L

Intensity [counts] 10

S

0.5

,S

x eq

0.4

L

xeq, i

0.3 0.2

L xeq, R =

0.1

S f(xS)

0.5

0.6

0.7

S xS

0.8

deS % 98

1.5

43.9 23.3

1.0

3.3

0.5

0 -3.3

0.0 0.9

0.0

1.0

5

10

15

20

25

30

2 Theta [o]

Fig. 3. Characterization of the solid phase behavior of System I. (A) SLE data, symbols experimental data, lines - polynomial approximation; (B) XRPD patterns for mixtures of crystals with different deS.22

A)

B) Intensity [counts] 10-4

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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3.0

de2,4 % 100

2.5

98

2.0 73

1.5

45 27

1.0

12

0.5

-11

0.0 5

10

15

20

25

30

35

2 Theta [°]


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Fig. 4. Characterization of the solid phase behavior of System II. (A) SLE data, hatched area corresponds to the region of two equilibrium states, red line – polymorph II, blue line – polymorph I; (B) XRPD patterns for mixtures of crystals with different de2,4.25 𝐿 Table.1. Coefficients of polynomial: 𝑥𝑒𝑞,𝑖 = 𝐚 (𝑥𝑖𝑆 )4 + 𝐛 (𝑥𝑖𝑆 )3 + 𝐜 (𝑥𝑖𝑆 )2 + 𝐝 (𝑥𝑖𝑆 ) + 𝐞, used to describe the SLE relationship.

𝐚 𝐿 𝑥𝑒𝑞,𝑆 𝐿 𝑥𝑒𝑞,𝑅

𝐛

𝐜

𝐝

𝐞

System I 0.00 0.00

0.00 − 0.0583 0.00 − 0.0618

1.314 − 0.0622 0.098 − 0.0382

System II

𝐿 0.00 − 0.0892 𝑥𝑒𝑞,2,4 𝐿 2.804 𝑥𝑒𝑞,1,3(*) − 1.378 𝐿 − 16.40 57.51 𝑥𝑒𝑞,1,3(**)

0.2402 − 0.2182 0.0727 − 1.953 0.548 − 0.0332 − 75.25 568.0 − 9.313

𝑆 *) in the range 𝑥2,4 = 0.48–0.72 (polymorph II) 𝑆 **) in the range 𝑥2,4 = 0.72–1.00 (polymorph I)

The SLE and XRPD measurements were found to be reproducible for both System I and System II.

4.2. Crystallization kinetics 4.2.1. System I The crystallization kinetics was measured for diastereoisomeric mixtures of S- and R-citalopram·(+)DTT with different compositions (section 3.2.4). As mentioned in section 3, the experiments were performed for the seeded and unseeded solutions, which was aimed to quantify the effect of seeding on the crystallization course. In the seeded crystallization, a small amount of seed crystals with 𝑑𝑒𝑆𝑆 = 4% (𝑥𝑆𝑆 = 0.52) was introduced into the solution prior to the start of the crystallization process (section 3.2.4). Typical experimental data in the form of the time-dependent concentration profiles of the target compound (S-citalopram·(+)DTT) in the mother liquor, 𝑥𝑆𝐿 (𝑡), and the corresponding diastereoisomeric excess, 𝑑𝑒𝑆𝐿 (𝑡), are illustrated in Figs 5A-5D. The 𝑑𝑒𝑆𝑆 (𝑡) curves can be used to support the process design, i.e., to find a trade-off between yield and rate of the process. It can be observed that the process can be accelerated by seeding the solutions (Fig. 5A, 5B). In case of the seeded crystallization, the concentration drop in the mother liquor is almost instantaneous, while it is delayed for the unseeded one. This indicates that nucleation is the rate limiting step of the operation. The kinetic rate enhances with increasing the absolute difference between the feed (initial) and equilibrium composition (driving force of crystallization, Eqs 7,8). This can be observed by comparing the courses 15 ACS Paragon Plus Environment


Page 16 of 26

of the unseeded crystallization exemplified by the curves presented in Fig. 5, e.g., 1 and 3 𝐿 in Fig. 5C (the same driving force, (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 ) = 2.3 x 10−2 , the same kinetics) and 1 in 𝐿 Fig. 5A (lower diving force (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 ) = 1.3 x 10−2 , slower kinetics). Each feed (initial)

concentration corresponds to

different equilibrium concentrations,

which is

characteristic for solid solution forming systems. This is illustrated by divergent conodes on the ternary solubility phase diagram (Fig. 2B). To describe quantitatively the observed trends in the experimental data, the kinetic model expressed by Eqs (1-14) was employed. Since there is only one crystalline phase (no polymorphs detected within the composition range investigated), NP was set equal to 1, and the index p in the model equations was not used. For the sake of simplicity, the volume shape factor was set kv = 1, the critical diameter of the nuclei was set Lcrit ~ 0. The model was solved using a standard numerical procedure for ordinary differential equations, which was implemented into an optimization routine, whose objective function (OF) was the sum of squared differences between the model simulations and the experimental kinetic profiles: 𝐿 𝑁 𝐿 𝑂𝐹 = ∑𝑁𝐶 𝑖 ∑𝑘 (𝑥𝑖,𝑘,𝑠𝑦𝑚 − 𝑥𝑖,𝑘,𝑒𝑥𝑝 )

2

(17)

𝐿 𝐿 where 𝑥𝑖,𝑘,𝑠𝑦𝑚 , 𝑥𝑖,𝑘,𝑒𝑥𝑝 are the liquid phase concentrations of the component i that were

simulated and measured for each data point k, respectively; NC is the number of the components (NC = 2); N is the number of the data points. A preliminary numerical study revealed that the crystallization kinetics of both seeded and unseeded crystallizations could be described by the same type of the kinetic equation (Eqs 7, 8), and that the nucleation rate for both S- and R-diastereoisomers could be expressed by the same nucleation rate coefficients, i.e., kB,i = kB, and the same empirical exponent ri = r, i.e., the order of the kinetic equation that determines the driving force dependence (Eq. 7). In such a case, the contribution of each diastereoisomer to the overall 𝐿 nucleation rate depended only on the individual driving force (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 ). Moreover, the

same exponent could be used to describe the driving force dependence of the crystal growth rate, i.e., qi = q (Eq. 8). Eventually, five model coefficients were estimated: kB, r, (Eq. 7), and kG,S, kG,R, q (Eq. 8), where the subscripts S and R denote the diastereoisomers S and R, respectively. It should be kept in mind that the kinetic parameters kB, kG,S, kG,R lump the contribution of the hydrodynamic conditions to the process rate. The exponents r and q were changed discretely using integer numbers in the range 1 - 10. The parameters were 16 ACS Paragon Plus Environment

Page 17 of 26

estimated by fitting the model solution to two sets of kinetic data in the form of timedependent concentration profiles of the target product, S, in the mother liquor, which were acquired during the course of the seeded and unseeded crystallizations. The estimation procedure was based on the minimization of OF in Eq. (17). The experimental data used for the estimation and the corresponding model simulations are depicted in Fig. 5A. The model parameters obtained are reported in Table 2. It is evident that the crystallization kinetics of S-diastereoisomer are almost half that of the Rdiastereoisomer (compare the values of kG,S and kG,R), which indicates the possibility of optimizing the crystallization efficiency by altering the process duration. In case of both nucleation and crystal growth, the second order kinetic equation (r = 2, q = 2) was found to be the most accurate in describing the kinetic rates. Next, the model along with the parameters determined was used to predict the process kinetics for all remaining mixtures with different compositions. Typical quality of the predictions is presented in Fig. 5C, where the time-dependent concentration profiles are shown, and in Fig. 5D, where the corresponding changes in 𝑑𝑒𝑆𝐿 are depicted.

A)

5.00

xSF,S = 0.75

System I

4.75

B)

85

70

1

2

4.00 3.75

65 60 55

3.50

50

3.25

45

0

10

20

30

40

t [h]

0

50

D)

16 14

F,S

xS

1 - 0.90 2 - 0.75 3 - 0.55

8 6

3

40

50

60

70

1

80

10

2

30

90

12

2

20

100

1

4

10

t [h]

2

70

deLS [%]

C)

2

75

deLS [%]

xLS x 102

4.25

1

80

1 - unseeded 2 - seeded

4.50

xLS x 102

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


60 50 40

3

30 20 10

0 0

10

20

30

40

t [h]

50

60

70

0

10

20

30

40

50

60

70

t [h]

Fig. 5. Course of crystallization in System I. A) Time-dependent concentration profiles in the mother liquor for the seeded and unseeded crystallizations – data used for the estimation of the model parameters; B) changes in 𝑑𝑒𝑆𝐿 corresponding to A); C) time-dependent concentration



Page 18 of 26

profiles of S-citalopram·(+)DTT in the mother liquor in the unseeded crystallization for different feed compositions, 𝑥𝑖𝐹,𝑆 is the solid-phase feed concentration; D) changes in 𝑑𝑒𝑆𝐿 corresponding to C). Symbols – experimental data, lines – simulations.

It can be observed that the model was efficient in reproducing the process kinetics for different diastereoisomeric excesses of the feed solutions. Therefore, it can potentially be used to predict and optimize the efficiency of multistage crystallization, in which the mixture composition, the equilibrium concentration, and the process kinetics are changed in each of subsequent steps. Table 2. Model parameters estimated for System I.

kB [min-1] x (𝜌 𝑆 )2 r [-] kG,S [m min-1] x (𝜌 𝑆 )2 kG,R [m min-1] x (𝜌 𝑆 )2 q [-] 6.5 x 10-2

2

1.34 x 10-8

2.57 x 10-8

2

3.2.2. System II The kinetic measurements and the determination of the model parameters for System II were performed using similar procedure to that described for System I. The crystallization kinetics were measured for the mixtures of two racemates 2,4 and 1,3. As mentioned in section 3, since the goal of the crystallization was to separate the racemates instead of the single mixture components, the quaternary mixture was considered as a pseudo-binary one that consisted of two components, i.e., the two racemates. To determine the impact of seeding on both the process kinetics and the phase behavior, the seeded and unseeded crystallizations were performed in parallel for mixtures with the feed composition in the range corresponding to the occurrence of dual equilibrium states (hatched area in Fig. 4A). The seeded crystallization was markedly faster than the unseeded one (Figs 6A, 6B), and the composition of seeds determined the course of the process and the equilibrium state (compare the curves 1,2,3 in Figs 6A and 6B). Therefore, the course of the crystallization and the occurrence of desired state could be imposed by addition of seeds with appropriate de. When no seeds were added, the equilibrium states were established randomly. As already mentioned, we attributed this behavior to the co-existence of different polymorphs, which both formed solid solutions, but with different solid phase compositions. The crystallization kinetics was slow for low values of diastereoisomeric excess of the racemate 2,4 and accelerated significantly with increase in that value (Figs 6C, 6D). This


Page 19 of 26 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


indicated a very strong dependence of crystallization kinetics on the driving force, 𝐿 (𝑥𝑖𝐿 − 𝑥𝑒𝑞,𝑖 ).

To describe the process kinetics, we a priori assumed the equality of the nucleation rate for both polymorphs, which was justified by randomness of the occurrence of the equilibrium states in unseeded crystallization within the critical composition range of the mixtures. This meant no preference towards the formation of a defined polymorph. The model equations for each polymorph consisted of the equations for the total and the individual component moments (Eqs 2 and 3 with NP = 2), the nucleation rate equations, and the crystal growth equations for the racemates 1,3 and 2,4. Both polymorphs differed in the equilibrium relationship, which was illustrated by the upward or downward sloping parts of the equilibrium curve depicted in Fig. 4A, and the kinetic parameters for the crystal growth rate. Eventually, the following set of parameters was estimated: kB, r (Eq. 7), kG,1,3,I, kG,1,3,II; kG,2,4,I kG,2,4,II, q (Eq. 8), where I and II are assigned to the polymorphs (Fig. 4A, Table 1). For the sake of simplicity, the volume shape factor was set kv = 1, the critical diameter of the nuclei was set Lcrit ~ 0, similarly as it was done for System I. The data set selected for the estimation of the model parameters consisted of two kinetic profiles acquired in the seeded crystallization illustrated in Fig. 6A, which yielded two different equilibrium concentrations corresponding to the formation of two polymorphs, i.e., the curve 2 to polymorph II, the curve 3 to polymorph I. The values of the determined parameters (kG,2,4, kG,1,3 in Table 3) indicate that in case of the polymorph I, the crystallization kinetics for the desired racemate 2,4 were slower than for the racemate 1,3, whereas the opposite held true for the polymorph II. This has to be taken into account while optimizing the process performance. The order of the nucleation rate was found to be very high, i.e., r = 8, which aroused from its strong dependence on the driving force of the process. This manifested itself by rapid reduction in the duration of the concentration plateau period with increasing the difference between the initial (feed) and final (equilibrium) concentrations (Figs A-D). The concentration plateau period is correlated with the nucleation period, in which crystal growth is inhibited due to slow nucleation rate. Next, the model was used to predict the unseeded crystallization for the feed solutions 𝑆 𝑆 𝑆 𝑆 with different 𝑑𝑒2,4 varied over the range 𝑑𝑒2,4 = - 20% (𝑥2,4 = 0.4) to 90% (𝑥2,4 = 0.95),

which was both inside and outside of multiple equilibrium states. The model was efficient in the whole range of the mixture composition investigated. 19 ACS Paragon Plus Environment


Table 3. Model parameters estimated for System II for the polymorphs I and II. kB x (𝝆𝑺 )8 [min-1]

r

A)

1.6

polymorph I

polymorph II

8

[m min-1]

[m min-1]

[m min-1]

[m min-1]

2.31 x 10-8

5.83 x 10-7

1.81 x 10-8

1.1 x 10-8

System II

B)

= 0.40

2

1.2 1.0

deS2,4 [%]

2

x 10

30

𝑆 Seeded 𝑥2,4 1 - 0.25 2 - 0.99 3 - Unseeded

3

2

40

1.4

1

q

[-] kG,2,4, I x (𝝆𝑺 )2 kG,1,3, I x (𝝆𝑺 )2 kG,2,4, II x (𝝆𝑺 )2 kG,1,3, II x (𝝆𝑺 )2 [-]

1.66 x 10-6

3

20

10 0.8

1 2

0.6 0

10

20

30

40

t [h]

50

70

80

0

D)

3.5

1 - 0.98 2 - 0.75 3 - 0.40

1

2.5

2

2.0

10

20

30

40

50

60

70

80

t [h]

3.0 2

0

60

1

𝐹,𝑆 𝑥2,4 = 0.40

1

1.5

3

1.0

100

1

80

2

deS2,4 [%]

C) x 10

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

Page 20 of 26

60 40

3

20 0

0.5 0

10

20

30

40

t [h]

50

60

70

0

10

20

30

40

50

60

70

t [h]

Fig.6. Course of the crystallization in System II. A) Time-dependent concentration profiles in the mother liquor for the seeded and unseeded crystallizations, the curve 2 (polymorph II) and the curve 3 (polymorph I) correspond to the data used for the estimation of the model parameters; B) 𝑆 changes in 𝑑𝑒2,4 corresponding to A); C) time-dependent concentration profiles in the mother liquor for the unseeded crystallization for different feed concentrations, 𝑥𝑖𝐹,𝑆 is the solid-phase 𝑆 feed concentration; D) changes in 𝑑𝑒2,4 corresponding to C). Symbols – experimental data, lines – simulations.

5.

Conclusions

The kinetics of crystallization in two solid solution forming systems, i.e., System I: binary mixtures of diastereoisomeric salts of citalopram with (+)-O,O′-di-p-toluoyl-D-tartaric acid and System II: pseudo-binary mixtures of stereoisomers of nafronyl oxalate, were 20 ACS Paragon Plus Environment

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analyzed and quantified. Both systems differed in phase behavior; in the former one the target compound was enriched in the mother liquor, whereas in the latter one in the crystalline phase. In System I the equilibrium states were established reproducibly, regardless of the composition of the mixture, whereas in System II multiple equilibrium states were established, which most probably was caused by the formation of polymorphs. In both systems seeded and unseeded crystallization was performed, which was aimed to determine the influence of seeding on the crystallization course. The addition of seeds accelerated significantly kinetics in both systems, which indicated that the nucleation was a rate limiting step. Moreover, in case of System II, the composition of seeds affected both the equilibrium concentrations and the crystallization rates. To describe the process quantitatively, a cooperative kinetic model was developed, which accounted for the formation of mixed crystals from nuclei of each component, and synergy in formation of crystals and their growth. Furthermore, the model accounted for the possibility of formation of polymorphs. The model was based on the moment method that was modified to account for miscibility in the solid phase. The presence of seeds was accounted for by the initial conditions, i.e., by specifying the seed crystals moments. The model parameters were estimated based on the kinetic data acquired in different crystallization modes: seeded versus unseeded (System I), or seeded with seeds of different compositions vs unseeded (System II). The nucleation rate in a given system could be described by the same type of kinetic equations, which was of the 2-nd order for System I, and as much as the 8-th order for System II, for which the nucleation rate enhanced rapidly with increasing the driving force of the process. The crystal growth rate was described by the 2-nd order equation for both System I and System II, though the kinetic parameters were different for each of the components in each of the systems. This indicated the possibility of altering the duration of crystallization to amend the separation efficiency. The model was efficient in predicting the kinetics of the seeded as well as unseeded crystallizations in both systems. Such an efficient mathematical tool is indispensable for the design of crystallization in solid solution forming systems, in which multistage operation is required to achieve proper purity of the target product. Since each of the stages can differ in crystallization kinetics, the model predictions can be used for guiding the choice of the operating conditions to optimize the process efficiency. This approach



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can potentially be generalized for other solid solution forming systems, after proper adjustment of the kinetic rate equations.

6.

Nomenclature Symbols B0 de G0 k kv m n NP p q r t x

Subscripts and superscripts

nucleation rate [s-1] diastereoisomeric excess [%] overall crystal growth rate [m s-1] kinetic coefficient [min-1 or mxmin-1] volume shape factor mass [g] moment order polymorph number polymorph empirical exponent empirical exponent time [min] mass fraction Greek symbols

s

eq F i j L

equilibrium conditions feed component index phase index liquid phase (mother liquid) p polymorph index S solid phase Sol solvent

solid density [kg m-3]

Acknowledgements Financial

support

of

this

work

by

National

Science

Center

(project

UMO2013/08/M/ST8/00982) is gratefully acknowledged.

Author information Corresponding Author Phone/fax: +48 178651853;/ +48 178543655; e-mail: [email protected] Notes The authors declare no competing financial interest.

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(3) Randolph, A. D.; Larson, M. A. in Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization. Academic Press, New York, 1988, 2nd Edition. (4) Lorenz, H.; Seidel-Morgenstern, A. Processes To Separate Enantiomers. Angew. Chem. Int. Ed. 2014, 53, 1218 – 1250. (5) Oonk, H.; Tjoa, K.; Brants, F.; Kroon, J. The Carvoxime System. Thermochim. Acta. 1977, 19, 161−171. (6) Chion, B.; Lajzerowicz, J.; Bordeaux, D.; Collet, A.; Jacques, J. Structural Aspects of Solid Solutions

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(14) Bredikhin, A. A.; Bredikhina, Z. A.; Zakharychev, D. V.; Gubaidullin, A. T.; Fayzullin, R. R. Chiral Drug timolol maleate as a Continuous Solid Solution: Thermochemical and Single Crystal X-ray Evidence. Cryst. Eng. Comm. 2012, 14, 648−655. (15) Bredikhin, A. A.; Bredikhina, Z. A.; Zakharychev, D. V. Crystallization of Chiral Compounds: Thermodynamical, Structural and Practical Aspects, Mendeleev Commun. 2012, 22, 171−180. (16) Li, Y.; Zhao, Y.; Zhang, Y. Solid Tryptophan as a Pseudoracemate: Physicochemical and Crystallographic Characterization. Chirality. 2015, 27, 88−94. (17) Rekis, T.; Be̅ rziņ š , A.; Orola, L.; Holczbauer, T.; Actiņš , A.; Seidel-Morgenstern. A.; Lorenz H. Single Enantiomer’s Urge to Crystallize in Centrosymmetric Space Groups: Solid Solutions of Phenylpiracetam. Cryst. Growth Des. 2017, 17, 1411−1418. (18) Lin, S.W.; Ng, K.M.; Wibowo, C. Synthesis of Crystallization Processes for Systems Involving Solid Solutions. Comput. Chem. Eng. 2008, 32, 956–970. (19) Temmel, E.; Müller, U.; Grawe, D.; Eilers, R.; Lorenz, H.; Seidel-Morgenstern, A. Equilibrium Model of a Continuous Crystallization Process for Separation of Substances Exhibiting Solid Solutions. Chem. Eng. Technol. 2012, 35, 980–985. (20) Temmel, E.; Wloch, S.; Müller, U.; Grawe, D.; Eilers, R.; Lorenz, H.; Seidel-Morgenstern, A. Separation of Systems Forming Solid Solutions Using Counter-current Crystallization. Chem. Eng. Sci. 2013, 104, 662–673. (21) Balawejder, M.; Galan, K.; Elsner, M. P.; Seidel-Morgenstern, A.; Piątkowski, W.; Antos, D. Multi-stage Crystallization for Resolution of Enantiomeric Mixtures in a Solid Solution Forming System. Chem. Eng. Sci. 2011, 66, 5638–5647. (22) Balawejder, M.; Kiwala, D.; Lorenz, H.; Seidel-Morgenstern, A.; Piątkowski, W.; Antos, D. Resolution of a Diasteromeric Salt of citalopram by Multistage Crystallization. Cryst. Growth Des. 2012, 12, 2557–2667. (23) Olbrycht, M.; Balawejder, M.; Matuła, K.; Piątkowski, W.; Antos, D. Multistage Crossand Counter-Current Flow Crystallization For Separation Of Racemic 2-methylbutanoic Acid. Ind. Eng. Chem. Res. 2014, 53, 15990−15999. (24) Kiwala, D.; Olbrycht, M.; Balawejder, M.; Piątkowski, W.; Seidel-Morgenstern, A.; Antos, D. Separation of Stereoisomeric Mixtures of Nafronyl as a Representative of Compounds Possessing Two Stereogenic Centers, by Coupling Crystallization, Diastereoisomeric Conversion, and Chromatography. Org. Process Res. Dev. 2016, 20, 615−625. 24 ACS Paragon Plus Environment

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For Table of Contents Use Only

A Cooperative Kinetic Model to Describe Crystallization in Solid Solution Forming Systems

Liquid

SLE

nafronyl oxalate stereoisomers B) 1.6 kinetics 2

te 1,3

racema

mass frac. x2,4Liquid x 10

2.5

x 10

A)

3.0

2

Maksymilian Olbrycht, Maciej Balawejder, Izabela Poplewska, Heike Lorenz, Andreas Seidel-Morgenstern, Wojciech Piątkowski, Dorota Antos

2.0

mass frac. xeq

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

Page 26 of 26

State 2

State 1

1.5 1.0

racemate 2,4

0.5 0.0

1.4

composition of seeds: x2,4Solid

1.2

1 - 0.25 2 - 0.99

1.0

State 1 0.8

State 2

0.6 0.5

0.6

0.7

0.8

Solid

mass frac. x2,4

0.9

1.0

0

10

20

30

40

50

60

70

80

time [h]

A) Two different equilibrium states were established in the crystallization course of pseudo-binary mixtures of nafronyl oxalate racemates: (2R,2'R), (2S,2'S) (racemate 1,3) and (2R,2'S), ( 2S,2'R) (racemate 2,4). B) The solution was seeded to alter the crystallization kinetics and to impose the desired equilibrium state.


A Cooperative Kinetic Model to Describe Crystallization in Solid

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