A Population Balance Model for Temperature Cycling-Enhanced

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A Population Balance Model for Temperature Cycling-Enhanced Deracemization Antonios A. Fytopoulos, Christos Xiouras, Michail E. Kavousanakis, Tom Van Gerven, Andreas G. Boudouvis, and Georgios D. Stefanidis Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b00706 • Publication Date (Web): 02 Oct 2018 Downloaded from http://pubs.acs.org on October 2, 2018

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Crystal Growth & Design

A Population Balance Model for Temperature Cycling-Enhanced Deracemization Antonios A. Fytopoulos1,2, Christos Xiouras1, Michail E. Kavousanakis2, Tom Van Gerven1, Andreas G. Boudouvis2 and Georgios D. Stefanidis1,⃰ 1

Process Engineering for Sustainable Systems (ProcESS), Department of Chemical Engineering

KU Leuven, Celestijnenlaan 200F, 3001 Leuven, Belgium 2

School of Chemical Engineering, National Technical University of Athens, Athens 15780,

Greece *

Correspondence to: Georgios D. Stefanidis

Tel: +32(0)16321007, Email: [email protected] KEYWORDS: Ostwald ripening; Temperature Cycling; Deracemization; Population Balance Modeling

ACS Paragon Plus Environment

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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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ABSTRACT

It has been recently observed that a suspension of conglomerate crystals of both chiralities in contact with a solution where racemization takes place can undergo symmetry breaking due to periodic fluctuations in temperature. The interplay between the mechanisms behind this process is still a matter of debate, although there is consensus that growth / dissolution of crystals and racemization in the solution phase play a crucial role. In this work, we present a Population Balance Model (PBM), based on temperature-size dependent solubility, which simulates a system of enantiomeric crystals of both chiralities subjected to Ostwald ripening under periodic temperature fluctuations, in the presence of a liquid-phase racemization reaction. Our simulations reveal that for a racemic system with small initial size asymmetries between the enantiomers complete deracemization is achieved due to temperature fluctuations, whereas isothermal Ostwald Ripening only leads to partial enantiomeric enrichment. This implies that size-temperature dependence of solubility is an essential mechanism in temperature cyclingenhanced deracemization. Although enantiopurity is achieved with only growth, dissolution and temperature cycling, other mechanisms, such as breakage, could also be included. While our model qualitatively reproduces many of the experimental observations in temperature cyclingenhanced deracemization, the autocatalytic nature of the process is not captured and, similarly to Viedma ripening, some sort of chiral feedback mechanism (e.g. agglomeration or secondary nucleation) seems to be required to fully describe the process dynamics. The results presented here offer further insight into temperature cycling-enhanced deracemization and help pinpoint the influence of the various mechanisms in the process.


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1. INTRODUCTION Molecular chirality has several implications in the pharmaceutical industry, as the two enantiomers often have different biological activities and usually only one enantiomer possesses the desired therapeutic effect (eutomer)1. For instance, ibuprofen, which is sold under the trade names Advil, Motrin etc., is a racemate (1:1 mixture of the two enantiomers R- and S-), but only the S- enantiomer is active. The presence of the R- enantiomer turns out to hinder the body’s ability to utilize the S-enantiomer, since the pharmacological effect takes place in 38 min, versus 12 min for the pure S-enantiomer2. Since the implications of chirality can become more severe, as it was the case with thalidomide, regulatory bodies (EMEA, FDA) now exercise pressure on pharmaceutical companies to produce enantiopure drugs and to treat undesired enantiomers as impurities3. Until 2011, many enantiopure blockbuster drugs resulted from “chiral switches” of already existing racemates, but nowadays most of the newly introduced drugs are manufactured as de novo single enantiomers4. Enantiopure compounds can be obtained either by direct asymmetric synthesis or most often by the physical separation of the target enantiomer from a racemic mixture by e.g. chromatography or preferential crystallization. The latter is a powerful and scalable approach, but works best when each enantiomer forms individual crystals (conglomerates) and still in many instances the counter enantiomer nucleates compromising the enantiomeric purity. An alternative approach to preferential crystallization is based on the pioneering work of Viedma, who discovered experimentally that a racemic mixture of conglomerate crystals of the achiral NaClO3 in contact with their saturated solution, subjected to attrition, can undergo complete deracemization5. Later, Noorduin et al.6 successfully applied this method to organic conglomerate-forming enantiomers that undergo solution-phase racemization. The evolution of


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the enantiomeric excess (ee) towards complete enantiopurity was attributed to a complex interplay between particle formation by breakage, Ostwald ripening, enantioselective agglomeration and racemization7 and models to simulate these processes were developed by different groups8,9,10,11,12. Recently, it was shown that besides grinding, periodic temperature fluctuations in a suspension can also lead to enantiopurity in a process that is now referred to as temperature cyclingenhanced deracemization13. More specifically, during high temperature periods, both enantiomeric crystals in contact with their saturated solution dissolve leading to an increase in the solution phase concentration for both enantiomers. Subsequently, temperature decreases creating supersaturation with respect to both enantiomers. Since the two enantiomeric crystal populations are never identical (deliberate or accidental differences in size or mass are always present), it is expected that the supersaturation for the favored enantiomer (in terms of initial conditions) is depleted faster by crystallization compared to the counter enantiomer. The faster depletion of molecules of the preferred enantiomer from the liquid phase is immediately counterbalanced by the fast racemization leading to conversion of the undesired enantiomer molecules to the preferred enantiomer. The process is repeated periodically until enantiomeric excess reaches 100%. It is noted that in most experiments, the cooling rate is kept deliberately low to avoid significant particle formation by secondary nucleation, even though new research shows that secondary nucleation occurs when higher cooling rates or when a breakage source is applied and also leads to deracemization14-15. Typical experimental features of the temperature cycling-enhanced deracemization process are: a) autocatalytic enantiomeric excess amplification, similar to Viedma ripening, b) absence of intense grinding (in most experiments only gentle stirring was used), c) slower deracemization at


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higher solids loading or when temperature cycling is stopped, d) faster deracemization when damped temperature cycles are applied. Since the process is relatively new, few models have been presented in the literature with a view to describe the process. Suwannasang et al.16 proposed a mathematical model based on differences in the crystal growth kinetics between the enantiomers, arising from the assumption that since the enantiomeric crystals nucleate at different supersaturation levels, they exhibit different crystalline perfection and thus a difference in growth rate activities. This model is able to reproduce qualitatively the increasing evolution of the enantiomeric excess during temperature cycling. A later work by Uchin et al.17 evolved this model by using Population Balances and continuous functions. In the meantime, Katsuno and Uwaha proposed a model18, which deals with a cluster size distribution and is based on the dissociation of chiral clusters into monomers and the incorporation of small chiral clusters to solids with the same handedness. Even if most of the aforementioned models are able to reproduce some of the experimentally observed phenomena, more research needs to be done in order to understand the possible submechanisms and the exact interplay among them. More specifically, a critical question is whether temperature cycling-enhanced deracemization can be rationalized via some of the mechanisms thought to drive Viedma ripening, i.e. Ostwald ripening, breakage, agglomeration and racemization. In 2011, Iggland and Mazzotti, in a landmark paper10, modeled Viedma Ripening

using

enantioselective

Population

Balances

agglomeration

and

and

concluded

racemization

are

that the

size-dependent essential

solubility,

components

for

deracemization. In a recent paper19 of the same group, size-temperature dependent solubility was mentioned as possible mechanism behind the temperature cycling-enhanced deracemization process. Conversely,

Suwannasang et

al.16 state that temperature cycling-enhanced


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deracemization is a ripening process, but contrary to Viedma Ripening and Ostwald Ripening, size dependence of solubility may not be the dominant mechanism in the process. Meanwhile, the effect of temperature fluctuations on Ostwald ripening is rather well-known in the literature. For example, Winzer and Emons 20, while examining Ostwald Ripening under the influence of temperature cycles of 2˚C with a frequency of two cycles per hour for

42

KCl

crystals, found experimentally that the dissolution kinetics of small crystals were 20 times greater than those in isothermal Ostwald Ripening. Thus, the mass transfer between small and large crystals in Ostwald ripening is greatly intensified with the application of temperature cycles. In the same research, they compare dissolution kinetics of

KCl (radioactive isotope)

42

small crystals under temperature cycles and they claim that the dissolution of small crystals depends on the characteristics of the initial Particle Size Distribution (PSD). These experimental results could indicate that ageing of suspensions under temperature fluctuations could take place, being based on size-temperature dependent solubility, which is the thermodynamic basis of nonisothermal Ripening. This process was also discussed elsewhere21 in the past using the name “Temperature Cycling”, which essentially means the form of ripening where temperature fluctuations occur. In isothermal Ostwald ripening, larger crystals grow at the expense of smaller ones due to the size-dependence of solubility, but when temperature fluctuations are applied, solubility of all crystals is largely affected, but still, for the smaller particles, size-dependence of solubility plays an important role. During heating some crystals partially and some completely dissolve and during cooling all of them grow, making the use of the term “Ostwald Ripening” instead of “Ripening” equivocal, since the larger crystals do not grow at the expense of the smaller crystals (competitive process). Instead, the presence of temperature cycles in the process affects the dissolution and growth of the crystals depending on their size, that is, small crystals


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mostly dissolve and big crystals grow preferentially. In order to make the process of ripening under temperature fluctuations faster, Nývlt et al.22 designed and patented a crystallizer, which consists of two interconnected vessels, each operating at different temperatures. Finally, the design of an analogous apparatus by Suwannasang et al.23 for deracemization purposes, where temperature fluctuations as low as ± 2°C where applied, shows that there might be a connection between temperature cycling-enhanced deracemization and what is called Ripening under temperature fluctuations. It is clear from all the aforementioned, that more research is needed in order to reveal the underlying mechanisms behind Temperature induced-Deracemization. In this paper, we present a detailed Population Balance Model (PBM) to simulate the evolution of the enantiomeric excess in a system of enantiomeric crystals of both chiralities subjected to ripening under periodic temperature fluctuations in the presence of a fast liquid-phase racemization reaction. While in Viedma deracemization, additional mechanisms such as enantioselective agglomeration and attrition driven secondary nucleation are believed to occur, their influence on temperature cycling-enhanced deracemization is currently less clear. For the aforementioned reason we chose to neglect such phenomena and focus mostly on the effect of temperature on the crystal ripening process. However, it should be mentioned, that such mechanisms are considered essential to model the autocatalytic effect observed in systems that undergo deracemization with initial mass imbalance as the only difference between the two populations and they are likely to influence temperature cycling-enhanced deracemization as well. Nonetheless, the main purpose here is to study the influence of non-isothermal Ostwald Ripening on the temperature cycling-enhanced deracemization process. In the first part of the paper, we simulate the case of isothermal ripening for chiral systems exhibiting small initial size asymmetries. Then, we extend the model to account for ripening under programmed temperature


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cycles. Using the extended model, we investigate the effects of the applied temperature profile, initial size imbalances and enantiomeric excess, suspension density and compound solubility on the deracemization process. In addition, the effect of particle breakage is also included and discussed. Collectively, the work fills some gaps in the understanding of chiral ripening processes and helps pinpoint the influence of Ostwald ripening on temperature cycling-enhanced deracemization.

2. BACKGROUND INFORMATION 2.1 Isothermal Conditions When solid particles of different sizes are dispersed in a (nearly) saturated solution at constant temperature, there is a tendency for dissolution of the smaller particles and growth of the larger ones in a process known as Ostwald ripening. The main driving force behind the process is believed to be the difference in the solubility of the smaller particles, as compared to that of the larger particles, which usually becomes pronounced for particles smaller than 1 µm. Since the state of Ostwald Ripening24 is ultimate, i.e. the final outcome is one big crystal, in the case of different enantiomers in the solid phase, where racemization takes place in the liquid phase, complete enantiopurity can be achieved as well. As regards the limitations of this process, what should be mentioned here is that experimentally it is not trivial to observe the ultimate state of Ostwald Ripening (i.e. only one large crystal of one chirality as a final outcome) in a reasonable time interval, as an asymptotic quasi-stationary growth regime is attained in long times (this is also predicted by the LSW theory). The interesting consequence of this result is that after long times the shape of the PSD does not change and deracemization is hindered. As Iggland and Mazzotti showed theoretically25 growth, dissolution and racemization can lead to the complete


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dissolution of one population of crystals if racemization reaction is fast, breakage does not occur, particle concentration is low and the initial PSDs differ sufficiently. The thermodynamic basis of Ostwald ripening derives from the Gibbs-Thomson equation that describes the dependence of the critical size as a function of the supersaturation in the solution, S. Above the critical size xc , new nuclei are stable and can grow. If the size of the nuclei is smaller than xc , then the nuclei dissolve. The critical size xc is defined for faceted crystals as26:

xc =

where S=

c c∞ (T)

2ka γVm 3kv RTlnS

=

α

(1)

lnS

is the supersaturation in solution, c is the solution concentration, c∞ (T) is the

equilibrium concentration of a solution in contact with an infinitely large crystal at a given absolute temperature T, ka and kv are the surface and volume shape factors respectively, γ is the surface tension (average for all faces), Vm is the molar volume, R is the universal gas constant and α is the capillary length, a parameter that takes into account all the aforementioned physical parameters of the crystals. In Eq. 1, it is assumed that no dissociation27 occurs and that the surface tension γ is independent of the concentration and particle size. Also, the fact that GibbsThomson effect could cease to be influential at extremely small sizes of crystals is neglected21. By solving Eq.1 for S and taking the first two terms of Taylor expansion, the supersaturation for a x-sized particle can be written as:

S=

= exp ≈1+ x x (T)

ci=D,L c∞

α

α

(2)

2.2 Non-Isothermal Conditions


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In this work, the main goal is to study the effects of Ostwald ripening under temperature fluctuations, as a possible mechanism behind temperature cycling-enhanced deracemization. In Fig. 1, the proposed mechanism for temperature cycling-enhanced deracemization is depicted. During Step 1, the temperature remains constant at the low level, Tl , and Isothermal Ostwald Ripening is expected to be the predominant mechanism. Larger crystals grow at the expense of the smaller ones and the critical size is that of the low temperature Tl . When the temperature increases (Step 2 in Fig.1), the supersaturation decreases and the critical size increases21. This way, mass from all crystals dissolves during the heating step, as their size is smaller than the critical size at higher temperatures. Hence, only partial or complete dissolution of crystals takes place. During dissolution, the difference between critical and mean size is decreasing. In that way, for a system with initial mean size differences for the two enantiomers, smaller crystals of both enantiomers dissolve faster than larger ones, but the total amount to be dissolved is constant (and depends on ∆Τ), i.e. for larger ∆Τ more solid will be dissolved. For a single ideal cycle, the amount of crystal mass that dissolves at the heating step is theoretically equal to the amount that recrystallizes at the cooling step. Racemization in solution verifies the conversion of those molecules in the liquid phase, keeping the solution racemic. At the end of this step, only the larger crystals of both chiralities remain in the solution. The proposed predominant force here is that of size-temperature dependent solubility19. In Step 3, temperature remains constant at higher value (Th ) and crystals with sizes larger than that of the (new - higher) critical size (at Th ) grow at the expense of crystals with sizes smaller than the critical size at that temperature, which dissolve. Isothermal Ostwald Ripening occurs again, but at higher temperature. During this step, large crystals become larger only due to sizedependent solubility.


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In the fourth (final) step, the temperature decreases gradually and this leads to a minimization of the critical size, which becomes small enough to let all the remaining crystals in the solution grow. Larger crystals are submitted to a higher driving force for growth, due to their lower solubility, which allows them to grow faster than their smaller counterparts. The racemization process in the solution verifies the change in chirality and an overall net flux towards the favored, in terms of initial conditions, enantiomer takes place, leading to enantiomeric enrichment. In most experimental studies, the cooling rate is selected as such to avoid nucleation, but for some compounds its occurrence cannot be excluded, especially for the higher cooling rates13. Hence, the proposed basis for temperature cycling-induced deracemization is a size-temperature dependent solubility19, which forms the thermodynamic basis of Non-Isothermal Ostwald Ripening28. The crystals dissolve and grow, because of the differences between their sizes compared to the critical size, which in our case depends on both concentration and temperature. Ergo, the differences in temperature “push and move” the critical size towards higher values (during heating) and towards lower values (during cooling). That way, the crystals partially dissolve while heating and all of them grow while cooling.


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Fig.1. Temperature Cycling process: in the end, complete deracemization is achieved. There are numerous research papers that have studied the effect of temperature variations on Ostwald Ripening29,30,31,32,33 and describe Ostwald ripening under non-isothermal conditions. In order to expand Ostwald Ripening to non-isothermal conditions, changes due to temperature variations should be implemented in growth rate, in critical size over time and a temperaturedependent Ostwald Ripening constant should be formulated. Details on the specific formulation can be found in Steinbach et al.

28

, where the growth rates, Ostwald Ripening constants and

critical sizes xc are expressed in terms of temperature, as it is summarized in Table 1. Table 1. The different growth rates G , Ostwald Ripening constants ε and critical radius xc that are applicable to Isothermal and Non-Isothermal Ostwald Ripening. Type of Process

Growth Rate G

Ostwald Ripening Constant ε

Process

ε ε xxc x2

2γVm 2 Dc∞ RT

Isothermal

ε µ0 ε q Vm 1 - x µ1 x2 4πµ1 x

2γVm 2 D0 c∞ (T) Q exp (- ) RT RT

Isothermal Ostwald Ripening NonIsothermal Ostwald

Temperature Cycling

Ripening * x=size of spherical particles, γ=interfacial energy, D=diffusion coefficient, µ0 =

zeroth moment = f, t)dx , µ1 = first moment = x fx, t)dx , q = Q=activation energy and xc = critical size =

!"

!# $

%&' ()*

,

and f the PSD of the population.


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3. Model Description The model used in this work is a PBM that incorporates growth, size-dependent solubility, and racemization in solution. In addition, breakage is incorporated, since in most of the experimental cases agitation takes place13. The model equations are solved using COMSOL Multiphysics® software (the direct UMFPACK solver), which is based on the Finite Element Method. None of the simulations in this work required computational time more than 10 min in a Windows work station with an i7 4510U at 2 GHz processor and 8 GB RAM.

3.1 Population Balance Model for Isothermal Conditions First, a Population Balance Model to describe growth, dissolution (Ostwald ripening) and breakage of two enantiomeric populations at isothermal conditions was developed based on the work of Iggland and Mazzotti25. Through the mass balances, the model also includes a first order racemization reaction (interconversion of the two enantiomers in the liquid phase). The dimensionless form of the population balance equations that describe the evolution of the particle size distribution (PSD) of enantiomer i=D,L are the following10: ∂fi y,τ) ∂τ

+

∂(G*i y)fi y,τ)) ∂y

=Kb y fi ε,τ) εβ gy,ε) dε - yβ fi y,τ) ∞

(3)

where fi represents the Particle Size Distribution (PSD) of the population i (in all simulations an initial normal distribution is assumed) , τ= t+t represents dimensionless time, where t is the time 0 in s and t0 =x0 /kg , kg is the growth rate constant, y= x+x0 represents the dimensionless size of crystals, with x0 =10-5 m being a scaling factor to allow for simplicity in the calculations and G*i y) the dimensionless growth rate of crystals (i=D,L). The dimensionless constant Kb


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represents breakage rate, while g is the daughter distribution, which describes the fragments of the daughter crystals produced by a breakage event of a particle of size ε, which produces two fragments of sizes (ε-y) and y and β is a parameter in the breakage rate function. The above PBM equations are coupled with the mass balances for the solutes: dSD dτ dSL dτ

where ν=

kv ρ c∞

dµD,3

= -ν = -ν

+Κr ,SL -SD - and

(4a)

+Κr ,SD -SL -

(4b)

dτ dµL,3 dτ

is the hold-up ratio, i.e. the ratio of the density in the solid and liquid phase, ρ is the

density of the crystal and Κr is the dimensionless first order racemization rate constant. Equations (5) describe the dynamics of the third moment of particle size distribution for population i: dµi,3 dτ

= 3x30 0 y2 G*i y)fi y,τ) dy ∞

(5)

Crystal growth is assumed to be surface integration-limited10 and the growth rate is given by: G*i y)=

ci c∞ (T)

-1-

a x0 y

(6)

The above equations are interconnected, i.e. the evolution of the PSD is calculated by equations 3, where G*i y) is calculated by equations 6, using the values for Si that equations 4 provide. The daughter distribution for the breakage of a particle of size ε and the formation of two particles with sizes y and ε-y, respectively, can be described by equation10: 2 2z+1)

gy,ε)=3y2 2z+1) 3 ε

ε3

2z

(y- 2 )

(7)


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Here, it is assumed that binary breakage occurs providing two fragments that are randomly distributed25 (z=0 in equation 7). The particle size-dependent breakage rate is assumed to follow a power law according to: by)=Kb yβ , (β=1)

(8)

The boundary and initial conditions are the following: f i y=0,τ)=0

(9)

f i y,τ=0)= f 0i y)

(10)

µi,3 τ=0)=µ0i,3 =x0 3 0 y3 f i 0 y)dy ∞

and

Si τ=0)=S0i =1

(11)

(12)

In all cases, the same initial scaled25 PSD was used. More details about the variables and the dimensionless formulation can be found elsewhere10,25. Finally, the solid state enantiomeric excess expressed as:

ee=

µD,3 -µL,3 µD,3 +µL,3

(13)

provides the necessary information about the progress of the resolution, where µD,3 is the third moment of the PSD of population D. This means that for enantiopure D, ee=1 and for enantiopure L, ee= -1. 3.2 Population Balance Model for Non-Isothermal Conditions


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In order to account for the effect of temperature fluctuations, the present model is modified. The modifications include, for simplicity reasons, the temperature dependence of solubility c∞ (T), which leads to different values of ν and the temperature dependence of the capillary length, α (equation 1). During the temperature fluctuations, the value of ν varies leading to changes in the supersaturation for the two enantiomers (equation 5), which in turn affects the driving force for growth/dissolution (equation 6). In addition, as solubility changes with temperature, the critical size is affected as well (equation 1). In this work, it is assumed that temperature changes slowly via infinitesimal steps, during which the solution concentration changes infinitely fast to match the new solubility, i.e. the system attains quasi-saturated states during heating and cooling. The temperature dependence of solubility is most often expressed empirically using a nth order polynomial21: c∞ T)=l+kT+pT2 +…

(15)

where usually a second order polynomial is used. In this model, a temperature dependent solubility of the form c∞ T)=l+kT will be used. For small temperature fluctuations, as the ones often applied in temperature cycling-enhanced deracemization, the dependence of solubility on temperature can be considered essentially linear. Since dissolution rates are much faster than growth rates21, it is likely that dissolution plays an important role in the evolution of the deracemization. Besides solubility and capillary length, which are considered in this model, racemization reaction rate constant, growth rate constant and surface tension also depend on temperature, however, they are not taken into account here. Finally, it should be mentioned that the experimentally observed induction time is not expected here, as the model takes into account initial differences in the PSDs of the populations (e.g. mean sizes etc.) and because of the lack of


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the autocatalytic effect in the model, which gives rise to asymptotic rather than sigmoidal kinetics. Therefore, Ostwald ripening process is fast from the beginning of the process. 3.3 Simulation Conditions The initial conditions and temperature programs that are applied in all simulations are presented in Tables 2-4. Table 2 shows the initial parameters for the isothermal Ostwald ripening simulations, while Tables 3 and 4 show the temperature programs and the initial parameters used in the non-isothermal simulations, respectively. In Tables 2 and 4, sr=y. 0D /y/ L0 quantifies the initial difference in the size between the enantiomers, while 01 quantifies the initial broadness of the PSD of each enantiomer (standard deviation of the initial normal PSD). Table 2. Initial parameters for the isothermal Ostwald ripening simulations #V1- #V6. All parameters used in this work are dimensionless.

Simulation

#V1

23 0

24

586,7 , 589,7

::8

;3

0.95), contrary to the isothermal case at the same conditions (dashed green line). It is clear that complete deracemization is powered mostly by the temperature fluctuations that lead to a temperature-concentration dependent critical size, rather than due to Isothermal Ostwald ripening process with a concentration dependent critical size and constant temperature, but this will be examined later as well. The solubility seems to be important as we can observe fluctuations in the rate of the ee evolution. As can be seen in Fig. 4 (b), where the influence of a single temperature cycle on the evolution of the ee is presented, during heating (τ=100-150) and cooling (τ=200-250), the ee increases more rapidly, while during isothermal steps (τ=0-100 & τ=150-200) the evolution is the same as in Isothermal Ostwald Ripening. Thus, the deviation from Ostwald ripening occurs due to the temperature fluctuations. During the heating period, smaller crystals of both chiralities dissolve completely and this leads to an increased enantiomeric excess, as shown in Fig. 1. In the extreme case, all the particles of the minor (in terms of initial conditions) population could dissolve during heating, leaving only a few particles of the major enantiomer that serve as seeds during the subsequent cooling period, essentially leading to enantiopurity in a single cycle. This process is not controllable experimentally. Additionally, the solute concentration follows the saturation concentration, which increases and decreases according to temperature fluctuations, while the concentration in solution is the same for both enantiomers, indicating that the racemization reaction helps


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maintain the liquid solution racemic as shown experimentally13. In conclusion, according to the proposed model the mean initial size affects the evolution of ee.

Fig.4. (a) (top) The evolution of ee for a TC program for 20% different initial mean size between the enantiomeric crystals (b) (bottom) The evolution of ee during the first temperature cycle. 4.2.3 The effect of breakage (Simulation #3) Additionally to simple growth-dissolution, the combined effect of breakage was also studied. Clues that breakage could take place during the process have been found from Suwannasang et al.34 and Uchin et al.17. In Simulation #3, a nonzero breakage rate constant was implemented, while keeping all other parameters the same as in Simulation #2. In Fig. 5, the result of Simulation #3 is plotted with a dashed green line. We can observe that breakage only (without


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agglomeration) hinders complete deracemization, a result similar to the one found in the past when modelling solid state enantio-enrichment without agglomeration25. This is because breakage leads to homogenization of the particle size for both the populations and, hence, minimizes the initial asymmetries among the enantiomeric crystal populations. Since the preferred enantiomer crystals have now a smaller size, their growth is hampered as compared to the case without breakage (Simulation #2) leading to slower deracemization. This has already been observed experimentally in the past as well by Winzer et al20. However, in real systems, when breakage is significant, particles under 50 µm will be formed and agglomeration cannot be avoided. If agglomeration occurs in an enantioselective manner it can produce an autocatalytic effect, enabling deracemization through Viedma ripening25. This means that even if we can decouple breakage in our model, when breakage is present in the experiment, agglomeration cannot be avoided in most the times.

Fig.5 Effect of particle breakage on deracemization using temperature cycles. 4.2.4 Deracemization is powered by temperature cycling (Simulation #4) In order to prove that complete deracemization is powered mainly by temperature, we performed a simulation, where the temperature variations stop for a time interval. In Simulation #4, TP2


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was applied, but for the time interval τ=500-2000 where the temperature was held constant at Tl , i.e. Ostwald Ripening conditions. The same initial conditions with Simulation #2 hold and thus there is a small initial mean size imbalance in favor of the D enantiomer. Fig. 6 shows the evolution of the ee against dimensionless time (red dashed – dotted line). It is obvious that when the temperature cycles stop (at τ=500), the critical size is no longer affected by temperature and the process evolves as a normal Ostwald Ripening process, that is, there is an evolution of the ee similar to that of the green dashed line. After τ=2000, temperature cycles start again and the ee evolves as in Simulation #2, similar to the blue solid line. This is expected as, when the temperature cycling stops and the critical size depends only on concentration changes, there occur both growth and dissolution, due to size-dependent solubility, and the process becomes much slower (isothermal Ostwald Ripening). It is obvious that the evolution of the ee after temperature cycling restoration depends on the time over which temperature cycling is absent and on the evolution of ee due to isothermal Ostwald Ripening. If the time of absence is small, the ee is expected to evolve in the same manner as before stopping temperature cycling. In this simulation, we can see that if we stop the fluctuations, enantiopurity is delayed and the system cannot achieve a pure state in the same time interval. Hence it is clear that ripening is strongly accelerated by temperature fluctuations, in accordance with experimental observations reported in the literature13.


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Fig.6. The evolution of enantiomeric excess for Temperature Cycling (blue solid line), for Ostwald Ripening (green dashed line) and for Temperature Cycling with interruption in Temperature Cycles (red dashed-dotted line) 4.2.5 The effect of initial total mass (Simulation #5) The effect of the initial solid mass was investigated. In this simulation, we compare two different temperature cycling processes with the same initial conditions, but different initial total mass of the two enantiomers. In both situations, the initial enantiomeric excess is zero. In Fig. 7, the results of Simulation #5 are plotted. As expected, systems with larger initial mass for both enantiomers deracemize slower (blue solid line – Simulation #2). Since the system moves from lower to higher temperature, dissolution and growth take place, but the amount to be dissolved or recrystallized is the same and depends on the selection of the higher and lower temperatures. As a result, the same amount of mass will be dissolved and recrystallized each time, even when the initial total solid mass differs, leading to the conclusion that for smaller initial total masses a larger percentage of the mass is involved in the deracemization process. That way, the remained amount to be deracemized is smaller and deracemization is achieved earlier. This is in agreement with the experimental results presented in literature13,35.

Fig.7. Effect of the initial solids loading on deracemization rate.


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4.2.6 The effect of initial ee (Simulation #6) Experiments show that the larger is the initial enantiomeric excess, the sooner enantiopurity is achieved. This is predicted by the model, as seen in Fig. 8. Simulation #6 was started with different initial enantiomeric excess and different particle sizes between the enantiomers (D particles are larger and in excess). It is shown that the higher the ee at τ=0, the sooner complete deracemization is achieved. However, starting with the same initial ee end no size imbalance did not lead to deracemization (not shown here). Even though precisely equal size or broadness of the PSD between the enantiomers is not likely to be achieved experimentally, the initial enantiomeric excess, as the only difference at the starting point, is still expected to lead to complete deracemization, as seen in the experiments35. This discrepancy between model and experimental results exists because in our model, growth and dissolution are not directly dependent on the initial enantiomeric excess, but are only affected by crystal size10. On the other hand, mechanisms that depend on both the size and particle concentration (such as agglomeration and secondary nucleation) are not included in the present model. In other words, a system that starts with only a difference between the masses of the two enantiomers (ee ≠ 0) cannot undergo deracemization using the present model. Therefore, it is likely that another mechanism could play a decisive role in the process. The arguments mentioned above could lead someone to speculate that agglomeration (or secondary nucleation) could also affect the outcome of the process. Since the process is already complicated, further investigation is needed on this topic.


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Fig. 8. The effect of initial enantiomeric excess on the time needed to achieve the final state. 4.2.7 The effect of damped cycles (Simulation #7) It was recently observed that the use of damped heating-cooling cycles can speed up the deracemization34. In Simulation #7, we use a damped temperature profile (TP3), as presented in Fig. 9 (a). The evolution of the enantiomeric excess compared to that of the undamped program TP2 is plotted in Fig. 9 (b) with a green dashed line. As can be seen, while initially the deracemization rate is similar, the system with the damped cycles moves slightly faster towards enantiopurity as the system becomes more enriched in the preferred enantiomer. In the case of the undamped temperature cycles, the amount of solid that dissolves is the same in each cycle. However, towards the end of the process, the system contains mostly crystals of the preferred enantiomer. Thus, maintaining the same, rather large, temperature swings induces somewhat more dissolution of the preferred enantiomer crystals, leading to a delay in the achievement of complete deracemization. This finding is in accordance with experimental results that also show acceleration by applying a damped temperature cycling profile, while the method also has lower energy requirements34.


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Fig. 9 (a) (top) The profile of damped temperature cycles (b) (bottom) The effect of damped TC program on ee evolution. 4.2.8 The effect of solubility on deracemization (Simulation #8) For a given substance, the relative change in solubility with temperature in a particular solvent is expected to substantially influence the deracemization process. Experimentally, one could study the effect of the solubility by carrying out deracemization experiments in different solvents, in which the solute may exhibit markedly different solubility behavior. However, in such experiments a change in the solvent would be accompanied by changes in the all other rate processes (e.g. racemization and growth/dissolution), preventing accurate conclusions on the effect of solubility. In our model, the effect of solubility alone can be easily decoupled, by changing the slope of the c∞ (T) function, while keeping all other parameters constant. In Fig. 10, the evolution of the ee is presented for a case where solubility increases faster with temperature


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(k=11 and l= -2015). As can be seen, the more soluble the substance, the faster the system achieves enantiopurity. This is because the same temperature fluctuations lead to a larger fraction of the total mass being involved in the dissolution-growth cycles. Thus, the slope of the solubility curve is expected to play a main role in the process36.

Fig. 10. The effect of solubility of the substance on deracemization kinetics. 5. CONCLUSIONS In this work we have presented a detailed model, based on Population Balances to simulate the evolution of the solid phase enantiomeric excess when a racemic population of conglomerate crystals is subjected to ripening in contact with a racemizing solution under the influence of temperature fluctuations. Our results have revealed that for a racemic system with initial size asymmetries, periodic temperature fluctuations can lead to fast and complete deracemization, whereas isothermal Ostwald ripening only leads to slow and partial enrichment. We therefore suggest that size-temperature dependence of solubility, which is the thermodynamic basis of non-isothermal Ostwald ripening, is an essential mechanism in the recently discovered process of temperature cycling-enhanced deracemization. The model qualitatively reproduces most of the experimental observations in the aforementioned process, such as the effects of solubility, solids loading and damped temperature cycles. Additionally, the effect of particle breakage was also


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included and discussed. However, the autocatalytic kinetic nature of the temperature cyclingenhanced deracemization process, observed in all experiments, was not sufficiently captured only by taking into account the temperature-size dependence of solubility. Similarly to Viedma ripening, some sort of chiral feedback mechanism (e.g. enanstioselective agglomeration or secondary nucleation) seems to be essential in order to fully describe the process dynamics. Nonetheless, the present model helps fill some gaps in the understanding of the influence of Ostwald ripening in temperature cycling-enhanced deracemization and can become a valuable design and optimization tool for deracemization processes. AUTHOR INFORMATION Corresponding Author * Email: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. ACKNOWLEDGEMENT C.X. acknowledges funding of a Ph.D. fellowship by the Research Foundation-Flanders (FWO). Nomenclature α = dimensionless capillary length [-]


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β = parameter in the breakage rate function [-] ci = solute concentration in the continuous phase [kg/m3 ] c∞ =bulk solubility in continuous phase [kg/m3 ] ee=enantiomeric excess [-] ε=dimensionless size [-] f=scaled Particle Size Distribution [m-3 ] gy,ε)=daughter distribution [-] G*i =dimensionless growth rate [-] γ=surface tension [J/m-2 ] kg = growth rate constant [m/s] Kb =dimensionless breakage rate [-] Kr =dimensionless racemization rate [-] ka =area shape factor [-] kv =volume shape factor [-]

k=dissolution coefficient [

l=dissolution constant [

kg ] m3 K

kg ] m3


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xc =critical size of the particle [m] µi,3 =third moment of the PSD [-] ν=hold up ratio [-] ρ=crystal density [kg/m3 ]

R=universal gas constant [

J ] mol K

σ=standard deviation of PSD [-] S=supersaturation of solute in continuous phase [-] T=absolute temperature [K] τ=dimensionless time [-] t=time [s] t0 =parameter for scaling of dimensionless time [s] Vm =molar volume [m3 /mol] x=particle size [m] x0 =scaling parameter for size [m] y=dimensionless size [-] z=parameter in daughter distribution function [-]


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For Table of Contents Use Only

Manuscript title: A Population Balance Model for Temperature Cycling-Enhanced Deracemization

Author List: Antonios A. Fytopoulos, Christos Xiouras, Michail E. Kavousanakis, Tom Van Gerven, Andreas G. Boudouvis and Georgios D. Stefanidis TOC Graphic:

Synopsis: In this work a possible interplay between the mechanisms that take place in Temperature Cycling – Enhanced Deracemization is proposed. A detailed Population Balance Model (PBM), based on temperature-size dependent solubility, has been formulated and shows for the first time that Ostwald ripening under periodic temperature fluctuations could be the basis behind this process.


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A Population Balance Model for Temperature Cycling-Enhanced

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