Spontaneous and Controlled Macroscopic Chiral Symmetry Breaking by Means of Crystallization

S S symmetry

Review Spontaneous and Controlled Macroscopic Chiral Symmetry Breaking by Means of Crystallization

Gérard Coquerel * and Marine Hoquante

SMS EA3233, Place Emile Blondel, University of Rouen Normandy, CEDEX, F-76821 Mont-Saint-Aignan, France; [email protected] * Correspondence: [email protected]

Received: 21 September 2020; Accepted: 26 October 2020; Published: 30 October 2020

Abstract: In this paper, macroscopic chiral symmetry breaking refers to as the process in which a mixture of enantiomers departs from 50–50 symmetry to favor one chirality, resulting in either a scalemic mixture or a pure enantiomer. In this domain, crystallization offers various possibilities, from the classical Viedma ripening or Temperature Cycle-Induced Deracemization to the famous Kondepudi experiment and then to so-called Preferential Enrichment. These processes, together with some variants, will be depicted in terms of thermodynamic pathways, departure from equilibrium and operating conditions. Influential parameters on the final state will be reviewed as well as the impact of kinetics of the R S equilibrium in solution on chiral symmetry breaking. How one can ⇔ control the outcome of symmetry breaking is examined. Several open questions are detailed and different interpretations are discussed.

Keywords: chirality; deracemization; preferential enrichment; thermodynamics; phase diagrams; kinetics

1. Context, Introduction

1.1. Chiral Discrimination between Pairs of Enantiomers in the Solid State Two behaviors of the R–S system regarding racemization need to be distinguished. In the ﬁrst case, the two enantiomers do not interconvert under the operating condition or in the time scale of the experiment. Therefore, the system is a symmetrical binary system where the two components have exactly the same thermodynamic properties (temperature, enthalpy and entropy of fusion, density, Cp versus T, etc.). This behavior is represented in Figure1A–D. In the second case, the two enantiomers racemize rapidly in the liquid state. There is a relationship of interdependence between the two components, so the system is actually a degenerated binary system, as depicted in Figure1E.

Symmetry 2020, 12, 1796; doi:10.3390/sym12111796 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1796 2 of 17 Symmetry 2020, 12, x FOR PEER REVIEW 2 of 18

Figure 1. Binary system between a pair of enantiomers showing different types of chiral discriminations Figure 1. Binary system between a pair of enantiomers showing different types of chiral in the solid state: (A–D) in the absence of racemization in the liquid state; (E) with racemization in the discriminations in the solid state: (A–D) in the absence of racemization in the liquid state; (E) with liquid state. In cases (A–C), the vertical dashed lines represent metastable racemic compounds. System racemization in the liquid state. In cases (A,B,C), the vertical dashed lines represent metastable E represents a conglomerate with fast racemization in the liquid state. racemic compounds. System E represents a conglomerate with fast racemization in the liquid state. There are several degrees of chiral discrimination in the solid state [1]. The top chiral discrimination There are several degrees of chiral discrimination in the solid state [1]. The top chiral occurs when every single crystal contains a single enantiomer, which is called conglomerate. This is discrimination occurs when every single crystal contains a single enantiomer, which is called often defined as “spontaneous resolution” (Figure1A). The same case can exist with a partial solid conglomerate. This is often defined as “spontaneous resolution” (Figure 1A). The same case can exist solution (Figure1B). By extension, there is also the possibility of a complete solid solution with a with a partial solid solution (Figure 1B). By extension, there is also the possibility of a complete solid maximum or minimum point where, in the full composition range, both enantiomers are randomly solution with a maximum or minimum point where, in the full composition range, both enantiomers distributed over the crystallographic sites. (Figure1C). In the latter case, if there is no miscibility gap are randomly distributed over the crystallographic sites. (Figure 1C). In the latter case, if there is no in the solid state, the chiral discrimination in this single solid phase is very poor. Then, there is the miscibility gap in the solid state, the chiral discrimination in this single solid phase is very poor. Then, important heterochiral recognition corresponding to the formation of a stable <1-1> stoichiometric there is the important heterochiral recognition corresponding to the formation of a stable <1-1> intermediate solid phase, called the “racemic compound” (Figure1D). The latter case is by far the most stoichiometric intermediate solid phase, called the “racemic compound” (Figure 1D). The latter case popular: it accounts for 90–95% of all derivatives of a given couple of enantiomers. Crystallographic is by far the most popular: it accounts for 90–95% of all derivatives of a given couple of enantiomers. surveys show that most of these racemic compounds crystallize in a centro-symmetric space group Crystallographic surveys show that most of these racemic compounds crystallize in a centro- (P21/c, P-1, Pbca, C2/c have the greatest occurrence). Nevertheless, there are several hundreds of symmetric space group (P21/c, P-1, Pbca, C2/c have the greatest occurrence). Nevertheless, there are kryptoracemic compounds (KRCs) in CSD version 2020. These KRCs account for ca. 1% of the racemic several hundreds of kryptoracemic compounds (KRCs) in CSD version 2020. These KRCs account for compounds. In those structures corresponding to chiral space groups (also named Sohncke space ca. 1% of the racemic compounds. In those structures corresponding to chiral space groups (also groups), the two enantiomers are in equal amount in the unit cell, but as independent molecules [2–5]. named Sohncke space groups), the two enantiomers are in equal amount in the unit cell, but as Therefore, Z’ is an even number. When Z’ > 2, other possibilities can arise such as anomalous independent molecules [2–5]. Therefore, Z’ is an even number. When Z’ > 2, other possibilities can conglomerates [6–8]. arise such as anomalous conglomerates [6–8]. When changing the temperature, the chiral discrimination can be slightly or even completely When changing the temperature, the chiral discrimination can be slightly or even completely altered. Indeed, from a racemic compound at low temperature, a stable conglomerate can be obtained altered. Indeed, from a racemic compound at low temperature, a stable conglomerate can be obtained at higher temperature through a three-phase peritectoid invariant [1]. The opposite situation is also at higher temperature through a three-phase peritectoid invariant [1]. The opposite situation is also well known. The three-phase invariant is then a eutectoid [9]. The switch from a racemic compound to well known. The three-phase invariant is then a eutectoid [9]. The switch from a racemic compound a conglomerate-forming system can also appear with the addition of a particular solvent or co-crystal to a conglomerate-forming system can also appear with the addition of a particular solvent or co- former or both [10]. The addition of crystal co-formers (isolated or in a mixture of solvents, counter-ions, crystal former or both [10]. The addition of crystal co-formers (isolated or in a mixture of solvents, co-crystal former, etc.) greatly increases the number of possibilities to explore in order to spot at least counter-ions, co-crystal former, etc.) greatly increases the number of possibilities to explore in order one conglomerate [11,12]. This increases the order of the system, which is no longer binary, but ternary, to spot at least one conglomerate [11,12]. This increases the order of the system, which is no longer quaternary, quinary, etc. [13]. High-throughput techniques are of great help to alleviate the amount of binary, but ternary, quaternary, quinary, etc. [13]. High-throughput techniques are of great help to work due to the overwhelming number of tests [14,15]. alleviate the amount of work due to the overwhelming number of tests [14,15]. A complete solid solution at high temperature does not prevent the existence of a large miscibility A complete solid solution at high temperature does not prevent the existence of a large gap in the solid state at low temperature [16]. The chiral discrimination in the solid state increases miscibility gap in the solid state at low temperature [16]. The chiral discrimination in the solid state continuously as the two symmetrical solvus curves become more apart and towards a low temperature. increases continuously as the two symmetrical solvus curves become more apart and towards a low We will see that (paragraph on Preferential Enrichment: PE hereafter), for some special cases, temperature. and for systems initially very far from equilibrium reproducible chiral symmetry breaking can be We will see that (paragraph on Preferential Enrichment: PE hereafter), for some special cases, observed. Conversely, departure from equilibrium can be detrimental to the chiral discrimination and for systems initially very far from equilibrium reproducible chiral symmetry breaking can be in the solid state. Fast cooling can lead to solid solutions without PE effects and even more severe observed. Conversely, departure from equilibrium can be detrimental to the chiral discrimination in the solid state. Fast cooling can lead to solid solutions without PE effects and even more severe

Symmetry 2020, 12, 1796 3 of 17 Symmetry 2020, 12, x FOR PEER REVIEW 3 of 18 coolingcooling (i.e.,(i.e., quenching)quenching) can can lead lead toto amorphousamorphous material material without without any any kind kind ofof macroscopic macroscopic chiral chiral recognitionrecognition [[17].17]. PartialPartial solid solid solutions solutions between between enantiomers enantiomers are are also also well well knownknown [5[5,18].,18]. Those cases cases are are intermediateintermediate between between conglomerates conglomerate withouts without solid solid solution solution and and with with complete complete solid solidsolution. solution. In some In systems,some systems, the crystal the crystal growths growths of the pureof the enantiomers pure enantiomers can lead can to peculiarlead to peculiar microstructures microstructure nameds lamellarnamed lamellar conglomerates, conglomerates which should, which notshould be confused not be confused with racemic with racemic compounds compounds [19]. [19]. 1.2. Equilibrium in Solution 1.2. Equilibrium in Solution In solution, we can consider two diﬀerent extreme situations: no racemization and fast In solution, we can consider two different extreme situations: no racemization and fast racemization [10]. In the former, the chirality of the chemical entity is blocked in the solid state racemization [10]. In the former, the chirality of the chemical entity is blocked in the solid state as as well as in the liquid state. The latter encompasses: (i) the loss of chirality in a solution such as well as in the liquid state. The latter encompasses: (i) the loss of chirality in a solution such as sodium sodium chlorate (“racemization” is instantaneous here); (ii) some atropisomers with a low energetic chlorate (“racemization” is instantaneous here); (ii) some atropisomers with a low energetic barrier barrier between the enantiomers in solution; (iii) enantiomers that can rapidly interconvert by the between the enantiomers in solution; (iii) enantiomers that can rapidly interconvert by the action of action of a catalyst (a base, an acid, an enzyme, etc.) (Figure2). a catalyst (a base, an acid, an enzyme, etc.) (Figure 2).

Solvent S

R FigureFigure 2. 2. Degenerated conglomerate-formingconglomerate-forming ternary system with fast racemization racemization in in the the liquid liquid phase.phase. The The vertical vertical plane plane contains contains the racemicthe racemic liquor from liquor saturated from saturated solution at solution a diﬀerent at temperaturea different totemperature the vertical to line the ofvertical the pure line solvent. of the pure All othersolvent. compositions All other compositions of the liquid of phase the liquid are simply phase notare accessible.simply not The accessible. green and The blue green tie lines,and blue respectively, tie lines, connectrespectively, the S enantiomerconnect the andS enantiomer the R enantiomer and the toR theenantiomer saturated to racemic the saturated solution racemic at a given solution temperature. at a given temperature.

2. Macroscopic Spontaneous Chiral Symmetry Breaking Induced by Crystallization 2. Macroscopic Spontaneous Chiral Symmetry Breaking Induced by Crystallization Deracemization Induced by a Flux of Energy Crossing the Suspension (DIFECS) Deracemization Induced by a Flux of Energy Crossing the Suspension (DIFECS) When a fast racemization takes place in solution under the same condition as the crystallization When a fast racemization takes place in solution under the same condition as the crystallization of a conglomerate, it is possible to observe a spontaneous macroscopic break in symmetry. of a conglomerate, it is possible to observe a spontaneous macroscopic break in symmetry. The The corresponding phase diagrams are no longer those displayed in Figure1A–D, as detailed corresponding phase diagrams are no longer those displayed in Figure 1A–D, as detailed in [20,21]. in [20,21]. Indeed, the phase diagrams are degenerated because the liquid phase can only contain an Indeed, the phase diagrams are degenerated because the liquid phase can only contain an equimolar equimolar amount of enantiomers (i.e., a racemic composition see Figures1E and2). Under those amount of enantiomers (i.e., a racemic composition see Figures 1E and 2). Under those circumstances, circumstances, any energy ﬂux passing through the suspension for long enough will lead to the any energy flux passing through the suspension for long enough will lead to the disappearance of disappearance of one population of homochiral crystals. A constant mechanical stress such as grinding one population of homochiral crystals. A constant mechanical stress such as grinding (known as (known as Viedma ripening) [22], numerous temperature cycles [23], long exposure to ultrasound [24], Viedma ripening) [22], numerous temperature cycles [23], long exposure to ultrasound [24], pressure pressure stress [25] or microwaves [26] are general methods enabling the evolution of the initial stress [25] or microwaves [26] are general methods enabling the evolution of the initial dual dual population of particles to a single population of crystals containing a single enantiomer only, population of particles to a single population of crystals containing a single enantiomer only, i.e., i.e., deracemization. Those methods operate rather close to thermodynamic equilibrium. deracemization. Those methods operate rather close to thermodynamic equilibrium. In Figure 3, the isotherm shows the main feature of the process. V stands for the solvent, while S and R are the two enantiomorphous chemical entities. Due to the fast racemization in solution or

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Symmetry 2020, 12, x FOR PEER REVIEW 4 of 18 In Figure3, the isotherm shows the main feature of the process. V stands for the solvent, while S andsimply R are the twoabsence enantiomorphous of chirality inchemical solution, entities. the attainable Due to thestates fast of racemization the system are in solution inside the orsimply triangle theS- absenceR- LSAT and of chirality along the in racemic solution, line the V attainable- LSAT. LSAT states is the of point the system representative are inside of the the triangle saturated S-R- racemic LSAT andsolution along the at racemicthat temperature. line V- LSAT Under.LSAT is strong the point enough representative continuous of attrition, the saturated the racemic initial suspension solution atrepresented that temperature. by point Under I evolves strong towards enough FR continuous or FS (Figure attrition, 3A,B). Simultaneously the initial suspension, the racemic represented mixture byof point solids I evolvesrepresented towards by point FR or M F Sevolves(Figure towards3A,B). Simultaneously, S or R, i.e., the thepure racemic enantiomers mixture. The of solidstie lines representedconnecting by the point constant M evolves saturated towards liquid S orLSAT R,, i.e.,the point the pure representative enantiomers. of Thethe overall tie lines synthetic connecting mixture the constantand the saturated point representative liquid LSAT, of the the point solid representative composition move of the from overall LSAT synthetic–I–M to mixtureLSAT–FS– andS or theLSAT point–FR–R. representativeIf the system of does the solid not contain composition any movechiral fromimpurity LSAT –I–Mand the to LinitialSAT–F Stwo–S or populations LSAT–FR–R. Ifof thecrystals system are doessymmetrical not contain in terms any chiral of Crystal impurity Size andDistribution the initial (CSD) two populationsand Growth ofRate crystals Dispersion are symmetrical (GRD), the inlast termsstage of of Crystal evolution Size Distribution, that is, crystals (CSD) of and pure Growth S or Rate else Dispersion pure R in (GRD), equilibrium the last with stage L ofSAT evolution,, is purely thatstochastic. is, crystals Usually, of pure the S orkinetics else pure of this R inspontaneous equilibrium evolution with LSAT are, is of purely the first stochastic. order. In Usually, other words, the kineticslogarithm of this of enantiomeric spontaneous evolutionexcess (e.e.) are versus of the time first is order. linear. In The other more words, the overall logarithm composition of enantiomeric departs excessfrom (e.e.)50–50 versus composition, time is linear.the faster The the more evolution the overall towards composition homochirality departs is. Thus, from 50–50it is an composition, auto-catalytic theprocess. faster the However, evolution if towardsthe system homochirality contains chiral is. Thus, impurities, it is an the auto-catalytic growth and process. dissolution However, rates if of the the systemtwo enantiomers contains chiral become impurities, different, the promoting growth and one dissolution enantiomer rates ove ofr the the other. two enantiomers The final evolution become of diViedmafferent, promotingripening can one be enantiomer directed by over using the a chiral other. impurity The final and evolution the e.e of. versus Viedma time ripening can also can evolve be directedlinearly by [27]. using a chiral impurity and the e.e. versus time can also evolve linearly [27].

FigureFigure 3. 3.Spontaneous Spontaneous deracemization deracemization by by Viedma Viedma ripening, ripening ultrasound,, ultrasound microwaves, microwaves towards towards R R enantiomerenantiomer (part (part (A ))(A and)) and towards towards S enantiomer S enantiomer (part (part (B)). (B Starting)). Starting from from a suspension a suspension represented represented by

pointby point I, composed I, composed of a doubly of a doubly saturated saturated solution solution LSAT and LSAT an and equimolar an equimolar mixture mixture of pure chiralof pure solids, chiral thesolids, system the will system spontaneously will spontaneously break its symmetry break by its an symmetry evolution towardsby an evolution LSAT–FR–R towards (represented) LSAT–F orR–R

conversely(represented) to a symmetrical or conversely system: to a symmetrical LSAT–FS–S (notsystem: represented). LSAT–FS–S F(notR and represented). FS represent F theR and two FS possible represent ﬁnalthe compositions two possible offinal the composition overall synthetics of the mixture overall after synthetic symmetry mixture breaking. after symmetry breaking.

Viedma ripening can be implemented directly during the synthesis; this method is called Viedma ripening can be implemented directly during the synthesis; this method is called asymmetric synthesis, involving dynamic enantioselective crystallization [28]. asymmetric synthesis, involving dynamic enantioselective crystallization [28]. For example, isoindolinones, a class of compounds used as core structures for pharmaceutical For example, isoindolinones, a class of compounds used as core structures for pharmaceutical applications, were resolved successfully using Viedma ripening by the group of Vlieg [29]. Indeed, applications, were resolved successfully using Viedma ripening by the group of Vlieg [29]. Indeed, isoindolinones 1–3 (Figure4) crystallize in a conglomerate-forming system and racemize quickly in isoindolinones 1–3 (Figure 4) crystallize in a conglomerate-forming system and racemize quickly in solution without a catalyst. solution without a catalyst.

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Figure 4. Isoindolinones that could be deracemized by Viedma ripening. Figure 4. Isoindolinones that could be deracemized by Viedma ripening.ripening.

RepetitiveRepetitive oscillations oscillations oscillations of of temperature of temperature temperature are are also are also able also ableto ableinduce to to induce deracemization induce deracemization deracemization in the system in in the the without system system limitingwithoutwithout scale. limiting limiting This scale. spontaneous scale. This This spontaneous process spontaneous is known process process as Temperature is is known known as Cycle-Induced as Temperature Temperature DeracemizationCycle Cycle-Induced-Induced (knownDeracemizationDeracemization as TCID) (known [23 (known]. The as temperature asTCID) TCID) [23]. [23]. gradient The The temperature temperature could be in gradient space gradient [30 could] orcould in be time bein [inspace31 space], or [30] even [30] or both orin intime [32 time]. Figure[31][31], or,5 orshows even even both the both corresponding [32]. [32]. Figure Figure 5 phase shows 5 shows diagram, the the corresponding corresponding which illustrates phase phase the diagram evolution diagram, which, of which the illustrates system illustrates when the the evolution of the system when the temperature of the whole system fluctuates between TL and TH, the theevolution temperature of the system of the whole when systemthe temperature fluctuates of between the whole TL systemand TH fluctuates, the low andbetween high T temperatures,L and TH, the respectively.lowlow and and high high Whentemperatures temperatures the e.e., ofrespectively., therespectively. solid phase When When strongly the the e.e. e.e. departsof ofthe the solid fromsolid phase zero,phase strongly it strongly is beneficial departs departs to from decrease from zero zero, , theit isit amplitude beneficialis beneficial ofto thetodecrease decrease temperature the the amplitude amplitude oscillation. of ofthe The the temperature damped temperature eff ectoscillation. oscillation. avoids wasting The The damped timedamped and effect effect energy avoids avoids by preventingwastingwasting time time the and dissolution and energy energy by of by preventing the preventing enantiomer the the dissolution in dissolution excess [ 33of]. ofthe the enantiomer enantiomer in inexcess excess [33] [33]. .

Figure 5. TCID experiment: the inset shows the racemic vertical section versus temperature M–V–T, Figure 5. TCID experiment: the inset shows the racemic vertical section versus temperature M–V–T, startingFigure 5. from TCID a suspensionexperiment: of the composition inset shows I, composedthe racemic of vertical a racemic section mixture versus M of temperature pure enantiomers M–V– inT, starting from a suspension of composition I, composed of a racemic mixture M of pure enantiomers equilibriumstarting from with a suspension their saturated of composition solution at I T, Lcomposed. Repetitive of thermala racemic oscillations mixture M of of the pure suspension enantiomers from in equilibrium with their saturated solution at TL. Repetitive thermal oscillations of the suspension TinL equilibriumto TH back and with forth their lead saturated to complete solution deracemization at TL. Repetitive (i.e., chiralthermal symmetry oscillations breaking). of the suspension Here, only from TL to TH back and forth lead to complete deracemization (i.e., chiral symmetry breaking). Here, thefrom evolution TL to TH toback R isand represented, forth lead but,to complete in the absence deracemizat of anyion chiral (i.e., bias, chiral the symmetry system can breaking). evolve to Here S as, well.onlyonly the The the evolution amplitude evolution to and Rto is R kinetics representedis represented of the, but temperature, but, in, thein the absence oscillationabsence of anyof impactany chiral chiralthe bias bias kinetics, the, the system ofsystem deracemization. can can evolve evolve to to S asS as well. well. The The amplitude amplitude and and kinetics kinetics of of the the temperature temperature oscillation oscillation impact impact the the kinetics kinetics of of Figurederacemization.deracemization.5 schematizes the principle of deracemization by TCID; only one spontaneous evolution is shown: towards the R enantiomer. Two isotherms are represented: one at TL and the other one at TH. WhenFigure theFigure initial 5 schematizes 5 schematizes suspension the the I principle is principle put back of of andderacemization deracemization forth from TbyL by toTCID; TTCID;H, itonly undergoesonly one one spontaneous spontaneous partial dissolution evolution evolution is shown: towards the R enantiomer. Two isotherms are represented: one at TL and the other one at andis shown: recrystallization towards the cycles. R enantiomer. Starting fromTwo aisotherms suspension are represented represented by: one point at T I,L theand overall the other synthetic one at TH. When the initial suspension I is put back and forth from TL to TH, it undergoes partial dissolution mixtureTH. When evolves the initial towards suspension FR/H at I T isH putand back FR/L atand TL. forthSimultaneously, from TL to T theH, it compositionundergoes partial of the soliddissolution phase evolvesandand recrystallization recrystallization towards the purecycles. cycles. enantiomer Starting Starting from R from (or a S; suspension a the suspension latter case represented represented is symmetrical by by point point to theI, theI, onethe overall representedoverall synthetic synthetic in mixture evolves towards FR/H at TH and FR/L at TL. Simultaneously, the composition of the solid phase Figuremixture5). evolves It is worth towards noting FR/H that at T thereH and is FR/L an at initial TL. Simultaneously period without, the a noticeable composition evolution of the solid in the phase e.e. ofevolvesevolves the solid tow tow phase.ardsards the However,the pure pure enantiomer enantiomer the CSD R and (orR (or S GRD; theS; the andlatter latter probably case case is symmetricalis other symmetrical solid-phase to tothe the one attributes, one represented represented change in in FigureFigure 5). 5It). isIt worthis worth noting noting that that there there is anis aninitial initial period period without without a noticeable a noticeable evolution evolution in inthe the e.e. e.e. of of

Symmetry 2020, 12, 1796x FOR PEER REVIEW 6 of 18 17 the solid phase. However, the CSD and GRD and probably other solid-phase attributes, change during that period. This first step ends with what is called the “take-off”, a colloquial expression for during that period. This first step ends with what is called the “take-off”, a colloquial expression for the significant macroscopic evolution of the system. Kinetics can have a rather odd aspect, e.g., the the significant macroscopic evolution of the system. Kinetics can have a rather odd aspect, e.g., the system can remain seated on the “racemic fence” for several days before the take off. This phenomenon system can remain seated on the “racemic fence” for several days before the take off. This illustrates the stochastic aspect of spontaneous symmetry breaking. After this first period, the system phenomenon illustrates the stochastic aspect of spontaneous symmetry breaking. After this first shows the classical sigmoid evolution of the enantiomeric excess (e.e.) of the solid phase vs. time, period, the system shows the classical sigmoid evolution of the enantiomeric excess (e.e.) of the solid which means that it possesses first-order kinetics. phase vs. time, which means that it possesses first-order kinetics. The temperature versus time profile must be tuned for the achievement of the deracemization The temperature versus time profile must be tuned for the achievement of the deracemization and for its productivity, together with the minimization of the chemical degradation, if there is any. and for its productivity, together with the minimization of the chemical degradation, if there is any. The variation in solubility versus temperature is, of course, an important factor, but the cooling rate The variation in solubility versus temperature is, of course, an important factor, but the cooling rate is also important for the generation of small nuclei via secondary nucleation. This phenomenon is also important for the generation of small nuclei via secondary nucleation. This phenomenon participates to the turnover of the particles of the two populations of crystals [34]. participates to the turnover of the particles of the two populations of crystals [34]. Deracemization using temperature fluctuations was demonstrated on a precursor of paclobutrazol, Deracemization using temperature fluctuations was demonstrated on a precursor of a molecule of interest because of its role as a plant growth inhibitor (Figure6)[ 31]. In this case, paclobutrazol, a molecule of interest because of its role as a plant growth inhibitor (Figure 6) [31]. In the temperature fluctuations range between 20 C and 25 C or 30 C and the racemization is induced this case, the temperature fluctuations range between◦ 20 ◦°C and 25◦ °C or 30 °C and the racemization by sodium hydroxide. is induced by sodium hydroxide.

Figure 6. 6.1-(4-chlorophenyl)-4,4-dimethyl-2-(1H-1,2,4-triazol-1-yl)pentan-3-one, 1-(4-chlorophenyl)-4,4-dimethyl-2-(1H-1,2,4-triazol-1-yl)pentan a- precursor3-one, a of precursor paclobutrazol, of whichpaclobutrazol served as, which a model served compound as a model for TCID. compound for TCID.

In addition to mechanical or thermal energy fluxesfluxes (see(see above),above), other energetic fluxesfluxes passing through thethe suspension suspension lead lead to tocomplete complete deracemization deracemization (Deracemization (Deracemization Induced Induced by a Flux by a of Flux Energy of EnergyCrossing Crossing the Suspension the Suspension (DIFECS). (DIFECS) For instance,. For instance, periodic p variationseriodic variations in pressure in pressure [25] or pressure [25] or pressureand temperature and temperature [35], microwaves [35], microwaves [26], and [26] photons, and photons for light-sensitive for light-sensitive molecules molecules [11] have [11] proved have provedto induce to induce complete complete chiral chiral symmetry symmetry breaking. breaking This. This is not is not a limitative a limitative list. list. On On top top of of this,this, these stimuli have agonist effects effects and thus can be cumulated to speed speed-up-up the macroscopic macroscopic chiral symmetry breaking [36] [36].. The common features of these processes are that they are operated somewhat close to thermodynamic equilibrium with a stochastic c characterharacter regarding the final final stage stages,s, R or S. They show firstfirst-order-order kinetics,kinetics, that is,is, an autocatalytic global behavior. This does not mean that the predominant predominant mechanisms are all the same. For instance, instance, the application of ultras ultrasoundound could be faster than at attritiontrition to induce complete deracemization deracemization;; nevertheless,nevertheless, the finalfinal crystals are bigger [[24]24].. The agonist effects effects of those those various various fluxes fluxes of energy of energy seem moreseem consistent more consistent with several—concomitant—possible with several—concomitant— pathways.possible pathways.Several mathematical Several mathematical models have models been have proposed been proposed that fit pretty that wellfit pretty with well the sigmoidwith the shapesigmoid of shapee.e. variation of e.e. variation in the solid in the versus solid time versus [37 –time41]. [37 DIFECS–41]. DIFECS has been has proven been toproven be suitable to be suitable for general for generalapplication, application provided, provided the two following the two conditions following are cond fulfilled:itions a are conglomerate fulfilled: a is conglomerate in equilibrium is with in equilibriuma doubly saturated with a doubly solution saturated and the chemicalsolution and entities the undergochemical rapidentities racemization undergo rapid in solution racemization if not ininstantaneous—e.g., solution if not instantaneous in the case— ofe.g. a loss, in the of asymmetry case of a loss such of asymmetry as NaClO3 andsuch NaBrO as NaClO3. 3 and NaBrO3. For instance, glutamic acid could be deracemized by microwave-assistedmicrowave-assisted temperature cycling with a much shorter process time compared toto conventionalconventional temperaturetemperature cyclescycles [[26]26].. Kondepudi’s experiment experiment [42 [42]] is anotheris another illustration illustration of spontaneous of spontaneous chiral macroscopic chiral macrosco symmetrypic symmetrybreaking by breaking using crystallization by using crystallization in a conglomerate-forming in a conglomerate system.-forming Figure system7 illustrates. Figure this 7 experiment. illustrates Typically,this experiment. a racemic Typically, solution a racemic is cooled solution down is with cooled given down kinetics with ingiven a stirred kinetics medium. in a stirred If the medium. system Ifis the able system to generate is able a to single generate nucleus a single only nucleus (the “Eve” only crystal)(the “Eve for” crystal) a sufficient for a period sufficient of time period and of if time the andstirring if the rate stirring and stirring rate and mode stirring [43] mode are adequately [43] are adequately tuned, numerous tuned, numerous offspring crystals offspring will crystals be created will be created by collision with the stirrer or the wall of the reactor and thus a fast secondary nucleation

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Symmetry 2020, 12, x FOR PEER REVIEW 7 of 18 by collision with the stirrer or the wall of the reactor and thus a fast secondary nucleation originates fromoriginates this “Eve” from crystal. this “ ThisEve” phenomenon crystal. This dropsphenomenon the supersaturation drops the supersaturation of the medium. of At the the medium. end of the At process,the end a homochiralof the process, population a homochiral of crystals population is generated, of crystals which is generated are all descendants, which are of all the descendants one which of nucleatedthe one ﬁrst.which Ideally, nucleated by repeating first. Ideally, numerous by repeating experiments, numerous the results experiments should, bethe a results random should series be of a ( random) and (+) series populations of (−) and of crystals(+) populations that are statisticallyof crystals that close are to statistically 50–50%. close to 50–50%. −

Figure 7. Kondepudi’s experiment: in the inset, the temperature versus composition pathway of Figure 7. Kondepudi’s experiment: in the inset, the temperature versus composition pathway of the the mother liquor is represented. This is the racemic section as the liquid phase cannot deviate mother liquor is represented. This is the racemic section as the liquid phase cannot deviate from from enantiomeric excess (e.e.) = 0. In this experiment, it is possible that I-K1-LSAT and I-K2-LSAT enantiomeric excess (e.e.) = 0. In this experiment, it is possible that I-K1-LSAT and I-K2-LSAT pathways pathways lead to the homochirality of the solid phase. By contrast, KM might correspond to an lead to the homochirality of the solid phase. By contrast, KM might correspond to an excessive excessive undercooling which leads to several quasi simultaneous primary nucleation events and thus undercooling which leads to several quasi simultaneous primary nucleation events and thus a a heterochiral final population of crystals. heterochiral final population of crystals. This process was first described by Kondepudi, who found that spontaneous symmetry breaking upon theThis crystallization process was first of sodium described chlorate by Kondepudi occurs in stirred, who found solutions, that spontaneous whereas static symmetry conditions breaking give anupon equal the amount crystallization of L and of D sodium crystals chlorate [44]. Kondepudi’s occurs in stirred experiment solutions is also, whereas applicable static toconditions the molten give an equal amount of L and D crystals [44]. Kondepudi’s experiment is also applicable to the molten state, as a supercooled melt of 1,10-binaphthyl could crystallize with a large enantiomeric excess when vigorouslystate, as a stirred supercooled [45]. melt of 1,1′-binaphthyl could crystallize with a large enantiomeric excess when vigorouslyPreferential stirred Enrichment [45]. (PE hereafter) also appears as a spontaneous macroscopic symmetry breakingPreferential phenomenon Enrichment [46]. It was(PE hereafter) discovered also and appears has been as developeda spontaneous for 25macroscopic years by Professor symmetry Ruibreaking Tamura phenomenon at Kyoto University. [46]. It was Initially discovered confined and has to been a series developed of non-racemizable for 25 years by organic Professor salts Rui (firstTamura generation), at Kyoto it has University. since been Initially extended confined to other to families a series of of compounds non-racemizable such as organicamino acids salts and (first chiralgeneration) pharmaceutical, it has since drugs been (second extended generation to other compounds,families of compounds hereafterconsidered such as amino non-racemizable acids and chiral chemicalpharmaceutical entities). It drugs is, in a (second way, the oppositegeneration of preferential compounds crystallization., hereafter considered Indeed, the non binary-racemizable system correspondschemical entities). to an intermediate It is, in a way state, the betweenopposite Figureof preferential1B,C (see crystallization. below for a discussion Indeed, the on binary the nature system ofcorresponds the solid phase to an close intermediate to racemic state composition), between Figure it is run 1B, withC (see a below considerable for a discussion supersaturation on the nature and theof phase the solid that phase is very close much to racemic enriched composition), (>90% e.e.) it is is the run mother with a liquor.considerable By contrast, supersaturation the solid phaseand the exhibitsphase athat poor is very and oppositemuch enriched deviation (>90% (ca e.e.) 4–5% is e.e.).the mother In opposition, liquor. By preferential contrast, the crystallization solid phase [exhibits47] is runa poor rather and close opposite to equilibrium deviation with (ca a 4 conglomerate-forming–5% e.e.). In opposition system, preferential (Figure 1crystallizationA) and the very [47] much is run enrichedrather close phase to is equilibrium the solid. The with e.e. a of conglomerate the mother liquor-forming is opposite system to (Figure that of 1A) the and solid the and very usually much remainsenriched lower phase than is 20%the solid. (this value The e.e. is merely of the formother the best liquor cases). is opposite Several conditionsto that of the have solid been and pointed usually outremains in relation lower to than the success 20% (this of PE:value is merely for the best cases). Several conditions have been pointed out in relation to the success of PE: (i) A large solubility difference between the racemic composition (poorly soluble) and the pure (i) A large solubility difference between the racemic composition (poorly soluble) and the pure enantiomer (much more soluble) is necessary. The alpha molar ratio, α = s( )/s(+) = s( )/s( ), enantiomer (much more soluble) is necessary. The alpha molar ratio, α = s(±±)/s(+) = s(±)/s(± −), is is thus very small (this constitutes another contrast with preferential crystallization for which α thus very small (this constitutes another contrast with preferential crystallization for which α is usually comprised between: √(2) and 2). Various analyses of racemic solutions in different solvents led to the surmised existence of solvated homochiral assemblies.

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is usually comprised between: √(2) and 2). Various analyses of racemic solutions in different solvents led to the surmised existence of solvated homochiral assemblies. (ii) For a globally racemic composition, the crystal structure permits a certain degree of disorder between homochiral chains and/or planes, even if single crystals obtained from poorly supersaturated racemic solution (e.g., point Ω1.2 in Figure8 representing a supersaturation C/CSAT = 1.2) could have their structures resolved by X-ray diffraction in centrosymmetric space groups such as P-1 (the most frequent for PE) or P21/c. However, crystals obtained under high supersaturation (i.e., from a clear solution represented by point Ω8 in Figure8) clearly reveal, by Second Harmonic Generation (SHG), homochiral domains. If this effect cannot be detected, PE experiments fail [48]. (iii) The first generation of compounds showing PE effect have all exhibited solid–solid transitions between various disordered phases. For the second generation of compounds showing PE effect in some cases, no such solid–solid transition could be detected. A solid–solid transition during PE does not appear anymore as a mandatory condition for its success. (iv) From the first solid crystallized, which can have a high degree of stacking faults, a selective dissolution of domains containing the same enantiomer as that in excess in the solution occurs. A unique, detailed study [49] has shown that this dissolution is actually concomitant to the Symmetryre-incorporation 2020, 12, x FOR PEER of the REVIEW opposition enantiomer. It is thus the exchange of opposite enantiomers8 of 18 that is likely to be a concerted process. This results in a clear enrichment of the mother liquor and, (ii) Forsimultaneously, a globally racemic a slight composition, enrichment ofthe the crystal solid phasestructure in the permits opposite a certain enantiomer degree At of the disorder end of betweenPE, the solid homochiral phase appears chains to beand/or composed planes of, Heterogeneouseven if single Nearly-Racemic crystals obtained Crystals from (HNRC). poorly supersaturatedThis is different racemic from a genuine solution solid (e.g. solution, point (i.e.,Ω1.2 mixedin Figure crystals) 8 representing where a random a supersaturation distribution C/Cof theSAT two= 1.2) enantiomeric could have moleculestheir structures is observed resolved over by the X- crystallographicray diffraction in sites. centrosymmetric In HNRC, there space are groupssome homochiral such as P-1 domains (the most of frequent sub-micron for toPE) micron or P2 sizes.1/c. However, crystals obtained under high (v) supersaturationThe HNRC could (i.e. remain, from kineticallya clear solution stable represented for months by without point Ω a8 return in Figure to stable 8) clearly equilibrium reveal, by if Secondthe system Harmonic remains Generation unstirred in(SHG) a quiescent, homochiral state. domains. If this effect cannot be detected, PE experiments fail [48].

Figure 8. Second generation of compound presenting Preferential Enrichment (PE) eﬀect (without Figure 8. Second generation of compound presenting Preferential Enrichment (PE) effect (without polymorphism), when left very far from equilibrium, e.g., solution Ω8 +. From the eightfold supersaturation,polymorphism), crystallization when left very takes far place from and equilibrium the solution, pointe.g., moves solution along Ω8 a +. trajectory From the schematized eightfold supersaturation, crystallization takes place and the solution point moves along a trajectory by the red curve. The ﬁnal evolution of the system is represented by the solution LF and solid K—with schematized by the red curve. The final evolution of the system is represented by the solution LF and opposite chirality—both aligned with point Ω8, representing the overall synthetic mixture. solid K—with opposite chirality—both aligned with point Ω8, representing the overall synthetic mixture.

(iii) The first generation of compounds showing PE effect have all exhibited solid–solid transitions between various disordered phases. For the second generation of compounds showing PE effect in some cases, no such solid–solid transition could be detected. A solid–solid transition during PE does not appear anymore as a mandatory condition for its success. (iv) From the first solid crystallized, which can have a high degree of stacking faults, a selective dissolution of domains containing the same enantiomer as that in excess in the solution occurs. A unique, detailed study [49] has shown that this dissolution is actually concomitant to the re- incorporation of the opposition enantiomer. It is thus the exchange of opposite enantiomers that is likely to be a concerted process. This results in a clear enrichment of the mother liquor and, simultaneously, a slight enrichment of the solid phase in the opposite enantiomer At the end of PE, the solid phase appears to be composed of Heterogeneous Nearly-Racemic Crystals (HNRC). This is different from a genuine solid solution (i.e., mixed crystals) where a random distribution of the two enantiomeric molecules is observed over the crystallographic sites. In HNRC, there are some homochiral domains of sub-micron to micron sizes. (v) The HNRC could remain kinetically stable for months without a return to stable equilibrium if the system remains unstirred in a quiescent state. It is important to mention that PE has to be run at very high initial supersaturation and the system should not undergo any mechanical stress. For instance, smooth stirring introduced by the slow motion of a rocking plate is enough to derail PE and to return to normal thermodynamics

Symmetry 2020, 12, 1796 9 of 17

It is important to mention that PE has to be run at very high initial supersaturation and the system should not undergo any mechanical stress. For instance, smooth stirring introduced by the slow motionSymmetry of2020 a, rocking12, x FOR platePEER REVIEW is enough to derail PE and to return to normal thermodynamics without9 of 18 macroscopic symmetry breaking. Moreover, if the initial supersaturation is not high enough (β > four, without macroscopic symmetry breaking. Moreover, if the initial supersaturation is not high enough six, eight or more (!)) PE is not observed. Thus, PE is clearly a far from equilibrium process. In Figure8, (β > four, six, eight or more (!)) PE is not observed. Thus, PE is clearly a far from equilibrium process. the red tie line connecting the composition of the mother liquor and the solid at the end of the PE In Figure 8, the red tie line connecting the composition of the mother liquor and the solid at the end intersects the racemic line M–LRAC, which is a clear violation of the thermodynamics of equilibrium. of the PE intersects the racemic line M–LRAC, which is a clear violation of the thermodynamics of The thermodynamics of equilibrium are represented by the black tie lines and, within their context, equilibrium. The thermodynamics of equilibrium are represented by the black tie lines and, within it is impossible to have a break in symmetry, i.e., to have a solution and a solid of opposite chirality for their context, it is impossible to have a break in symmetry, i.e., to have a solution and a solid of a long period of time. When the macroscopic evolution of the system is over—this can take several opposite chirality for a long period of time. When the macroscopic evolution of the system is over— days—the composition of the mother liquor (e.g., +95% e.e.) and that of the solid (e.g., ca. 4–5% e.e.) this can take several days—the composition of the mother liquor (e.g., +95% e.e.) and that −of the solid can remain unchanged for months, maybe more, in a stagnant medium. (e.g., ca. −4–5% e.e.) can remain unchanged for months, maybe more, in a stagnant medium. An eﬃcient Preferential Enrichment phenomenon could be observed for the (DL)-phenylalanine An efficient Preferential Enrichment phenomenon could be observed for the (DL)-phenylalanine and fumaric acid co-crystal (Figure9)[50]. and fumaric acid co-crystal (Figure 9) [50].

Figure 9. 9.Cocrystal Cocrystal of (DL)-phenylalanineof (DL)-phenylalanine and fumaricand fumaric acid, a system acid, a that system exhibits that Preferential exhibits Enrichment. Preferential Enrichment. There is an interesting analogy between Lamellar conglomerates [19] and fluctuations in enantiomericThere is composition an interesting around analogy the racemic between composition Lamellar observed conglomerates in PE. On[19] the and one fluctuations hand, lamellar in conglomeratesenantiomeric composition correspond around to the stacking the racemic of homochiral composition domains observed [51 in]. APE. particle, On the crystallizedone hand, lamellar from a racemicconglomerates solution, correspond could look to very the stacking much like of a homochiral nice, single domains crystal, but [51] could. A particle, actually crystallized be constituted from by a theracemic alternation solution, of homochiralcould look domains.very much At like the a interface nice, single of opposite crystal, domains,but could a actually 2D racemic be constituted compound isby formed, the alternation but, for unclear of homochiral reasons, domains. this packing At does the interface not expand of in opposite the third domains, direction. a Quite2D racemic often, thecompound Flack parameter is formed [,52 but] reveals, for unclear trouble reason in thes, this absolute packing configuration does not expand assignment in the ofthird the direction. molecule andQuite the often, non-linear the Flack optics parameter of the powder [52] reveals show trouble an enhanced in the absolute SHG eff ectconfiguration compared assignment to that of a singleof the enantiomermolecule and with the the non same-linear CSD. optics The of global the powder composition show is an thus enhanced quasi-racemic, SHG effect but compared this is not actuallyto that of a singlea single crystal. enantiomer On the with other the hand, same when CSD. operatingThe global PE composition with large supersaturationis thus quasi-racemic, and under but this stagnant is not conditionsactually a single (and thereforecrystal. On far the from other equilibrium hand, when thermodynamics), operating PE with there large aresupersaturation also fluctuations and under in the compositionstagnant conditions of solid (and particles. therefore This far could from also equilibrium be revealed thermodynamics), by the SHG eff ect.there Thus, are also the fluctuations PE effect is linkedin the composition to the formation of solid of Heterogeneous particles. This Nearly-Racemiccould also be revealed Crystals by (HNRC) the SHG or, effect. in other Thus, words, the PE racemic effect compoundsis linked to the with formation local fluctuations of Heterogeneous in their enantiomeric Nearly-Racemic composition. Crystals Racemic(HNRC) crystalsor, in other that words do not, displayracemic thecom possibilitypounds with of local local enantiomeric fluctuations fluctuationin their enantiomeric do not exhibit composition. any PE eff Racemicects. crystals that do notStirring display the the crystallizing possibility suspensionof local enantiomeric curbs the fluctuationsfluctuation do in itsnot composition exhibit any andPE effect totallys. inhibits PE. LocalStirring deviations the crystallizing in the composition suspension of curbs the stagnantthe fluctuations mother in liquor its comp in theosition vicinity and oftotally the growing inhibits surfacesPE. Local are deviations the driving in the forces composition of those phenomena. of the stagnant For lamellarmother liquor conglomerates, in the vicinity when of an the S crystalgrowing is growing,surfaces are the the R enantiomer driving force iss overrepresented of those phenomena. in the For neighboring lamellar conglomerates, solution. The heteronucleation when an S crystal of Ris ongrowing, top of the SR crystalenantiomer is more is overrepresented likely as preferential in the crystallization neighboring solution. proceeds The towards heteronucleation the end of a of run R (seeon top Figure of the 10 Sand crystal caption). is more The likely resulting as preferential epitaxy is crystallization a sort of regulatory proceeds phenomenon towards the that end diminishes of a run the(see entrainmentFigure 10 and e ffcaption).ect, i.e., The drops resulting the magnitude epitaxy is of a sort the transientof regulatory symmetry phenomenon breaking. that diminishes One study hasthe entrainment shown that theseeffect, locali.e., drops fluctuations the magnitude could be of amplifiedthe transient at asymmetry macroscopic breaking. scale One in the study mother has liquorshown [ 53that]. Inthese the caselocal of fluctuations PE, the opposite could ebeffect amplified occurs: theat a incorporation macroscopic ofscale the minorin the mother enantiomer liquor in the[53] mother. In the liquorcase of around PE, the the opposite crystal effect and the occurs liberation: the incorporation of the minor enantiomerof the minor in enantiomer the solid (slightly in the inmother default liquor in the around solid) the constitute crystal anand amplification the liberation of of local the dissymmetryminor enantiomer (see Figurein the solid10 and (slightly caption). in default in the solid) constitute an amplification of local dissymmetry (see Figure 10 and caption). One can perceive this phenomenon as another type of ripening resulting from an initial, far from equilibrium crystallization.

Symmetry 2020, 12, 1796 10 of 17

One can perceive this phenomenon as another type of ripening resulting from an initial, far from Symmetryequilibrium 2020, 12 crystallization., x FOR PEER REVIEW 10 of 18

FigureFigure 10. 10. SchematicSchematic representation representation of of the the mechanisms mechanisms of of formation formation of of heterochiral heterochiral domains domains in in single single particlesparticles in in PE PE and and Preferential Preferential Crystallization Crystallization (PC (PC).). Left: Left: as asexplained explained in conditions in conditions for forthe thesuccess success of PE,of PE,solvated solvated homochiral homochiral associations associations are are likely likely to toexist, exist, and and are are symbolized symbolized as as vertical vertical redundant redundant lettersletters—S—S or or R. R. During During the the process, process, even even when when the the supersaturation supersaturation becomes becomes fairly fairly weak, weak, the the minor minor enantiomerenantiomer in in solution solution can can substitute substitute for for the the minor minor enantiomer enantiomer in in the the defective defective solid solid state. state. Thus, Thus, there there isis an an amplific amplificationation of the e.e. in the solid andand inin thethe mothermother liquor.liquor. Right: Right: focus focus of of the the crystal crystal growth growth of ofthe the S S enantiomer enantiomer during during PC, PC, the the mother mother liquorliquor isis depleteddepleted in thethe S enantiomer in in the the vicinity vicinity of of the the singlesingle crystal. crystal. As As R R is is supersaturated, supersaturated, it it is is possible possible for for this this enantiomer enantiomer to to heteronucleate heteronucleate on on top top of of its its antipodeantipode by by means means of of an an epitaxy. epitaxy. In In both both cases cases PE PE and and PC PC,, the the resulting resulting particles particles are are constituted constituted by by homochiralhomochiral domains: domains: sub sub-micron-micron to to several several microns microns size size in in PE PE and and up up to to hundreds hundreds of of microns microns in in PC. PC. One of the deep-seated reasons for those effects is the existence of an exact match between the One of the deep-seated reasons for those effects is the existence of an exact match between the crystal lattices of the enantiomorphous components (i.e., the Friedel and Royer conditions for the crystal lattices of the enantiomorphous components (i.e., the Friedel and Royer conditions for the junction of the two crystal lattices are perfectly fulfilled) [54]. junction of the two crystal lattices are perfectly fulfilled) [54]. 3. Control of Macroscopic Chiral Symmetry Breaking by Means of Crystallization 3. Control of Macroscopic Chiral Symmetry Breaking by Means of Crystallization It is possible to orientate the symmetry breaking towards a desired enantiomer by using several It is possible to orientate the symmetry breaking towards a desired enantiomer by using several robust methods. For instance, Deracemization Induced by a Flux of Energy Crossing the Suspension robust methods. For instance, Deracemization Induced by a Flux of Energy Crossing the Suspension (DIFECS) could lead to the eutomer (the desire enantiomer) by adding a small investment prior to (DIFECS) could lead to the eutomer (the desire enantiomer) by adding a small investment prior to the beginning of the process [55]. For example, only a small percentage of e.e. (+) in the solid state the beginning of the process [55]. For example, only a small percentage of e.e. (+) in the solid state is is sufficient to conduct the complete deracemization by using Viedma ripening, TCID, ultrasound, sufficient to conduct the complete deracemization by using Viedma ripening, TCID, ultrasound, microwaves, etc., towards the (+) enantiomer. This statement is valid if there is no chiral impurity in microwaves, etc., towards the (+) enantiomer. This statement is valid if there is no chiral impurity in the medium that could overcompensate the initial bias introduced purposely [56], that is, without this the medium that could overcompensate the initial bias introduced purposely [56], that is, without imbalance, the system would have a stochastic behavior. This is illustrated by the Viedma ripening of this imbalance, the system would have a stochastic behavior. This is illustrated by the Viedma sodium chlorate (used as received; see Figure 11). ripening of sodium chlorate (used as received; see Figure 11).

Symmetry 2020, 12, 1796 11 of 17 Symmetry 2020, 12, x FOR PEER REVIEW 11 of 18

FigureFigure 11. ImpurityImpurity effect effect on on the thehandedness handedness of the of attrition-enhanced the attrition-enhanced deracemization deracemization in a non-recrystallized in a non- recrystallizedbatch of sodium batch chlorate of sodium [57]. After chlorate a simple [57] recrystallization. After a simple in water, recrystallization this effect disappeared. in water, this effect disappeared. The evolution is systematically towards the same chirality even if an initial investment in the counterThe enantiomerevolution is issystematically performed. One towards experiment the same shows chirality an evolution even if an of initial up to investment 70% e.e. before in the a counterturn back. enantiomer A simple is recrystallization performed. One ofexperiment the initial shows racemic an mixtureevolution is of enough up to 70% to remove e.e. before that a e turnffect. back.It is likely A simple that arecrystallization small (maybe minute) of the amountinitial racemic of a chiral mixtu impurityre is enough “pushes” to remove the system that towardseffect. It the is likelysame enantiomers.that a small (maybe This phenomenon minute) amount has beenof a chiral observed impurity with “ organicpushes compounds” the system[ towards58]. Dissymmetry the same enantiomers.in crystal size This distributions phenomenon is also has able been to observed systematically with directorganic the compounds symmetry [58] breaking. Dissymmetry towards thein crystalpopulation size of distributi biggerons crystals is also (“big able is beautiful”). to systematically A study direct has shown the symmetry that there breaking is actually towards a balance the populationbetween the of initial bigger e.e. crystals and the (“big dissymmetry is beautiful” of).crystal A study size has distributions shown that there [59]. Biggeris actually crystals a balance of (+) betweencan, for instance,the initial compensate e.e. and the a slightdissymmetry initial excess of crystal in small size ( distributions) crystals [60 [5]. 9]. Bigger crystals of (+) − can, forKondepudi’s instance, compensate experiment, a seededslight initial with veryexcess pure in small enantiomer (−) crystals prior [60] to any. primary nucleation of eitherKondepudi enantiomer,’s experiment is equivalent, seeded to preferential with very crystallization.pure enantiomer A prior detailed to any analysis primary of thenucleation process of is eithergiven enantiomer elsewhere with, is equivalent differentprotocols to preferential for the crystallization. inoculation of A seeds detailed and analysis temperature of the profiles process [47 is]. givenThe symmetry elsewhere of with the systemdifferent is protocols purposely for broken the inoculation by the seeding: of seeds if theand solid temperature is the (+ )profiles enantiomer, [47]. Thethe mothersymmetry liquor of the evolves system towards is purposely an excess broken of by ( the) in seeding: the absence if the of solid racemization. is the (+) enantiomer, This induced the − mothersymmetry liquor breaking evolves lasts towards for some an excess minutes of (− to) in some the absence hours. Theof racemization. fine enantiopure This induced inoculated symmetry crystals breakinglead to stereoselective lasts for some growth minutes and to secondary some hours. nucleation; The fine during enantiopure this period, inoculated the counter crystals enantiomer lead to stereoselectiveremains in the growth supersaturated and secondary solution. nucleation; If the systemduring this is left period, for too the long, counter the enantiomer second enantiomer remains instarts the tosupersaturated spontaneously solution. crystallize If the so system that the is ultimate left for too evolution long, the of second the system enantiomer is a mixture starts to of spontaneouslycrystallized enantiomers crystallize in so equilibrium that the ultimate with a doublyevolution saturated of the racemicsystem solution.is a mixtu Ifre fast of racemizationcrystallized enantiomerstakes place in equilibrium the system with (or in a doubly the absence saturated of chirality racemic insolution. solution), If fast the racemization mother liquor takes cannot place indeviate the system from 0%(or e.e,in the as illustratedabsence of bychirality Figure in2). solution) In that case,, the themother preferential liquor cannot crystallization deviate from receives 0% e.eanother, as illustrated name: Second-Order by Figure 2). AsymmetricIn that case, Transformationthe preferential (SOAT;crystallization represented receive ins Figureanother 12 name:)[61]. SecondThis elegant-Order process Asymmetric could be Transformationtwo orders of magnitude (SOAT; morerepresented productive in thanFigure any 1 variant2) [61]. of This DIFECS elegant [62]. processSupersaturation could be has two to be orders kept within of magnitude reasonable more limits productive so that crystal than growth any variantand secondary of DIFECS nucleation [62]. Supersaturationremain stereoselective. has to be kept within reasonable limits so that crystal growth and secondary nucleation remain stereoselective.

Symmetry 2020, 12, 1796 12 of 17 Symmetry 2020, 12, x FOR PEER REVIEW 12 of 18 Symmetry 2020, 12, x FOR PEER REVIEW 12 of 18

Figure 12. 12. SecondSecond-Order-Order Asymmetric Asymmetric Transformation Transformation (SOAT) experiment: the the inset shows the Figure 12. Second-Order Asymmetric Transformation (SOAT) experiment: the inset shows the temperature versusversus compositioncomposition pathwaypathway ofof the mother liquor. Point I does not belong to the isopleth temperature versus composition pathway of the mother liquor. Point I does not belong to the isopleth section represented inin thethe inset,inset, asas thethe seedseed crystalscrystals areare enantiopure.enantiopure. section represented in the inset, as the seed crystals are enantiopure. In 1979, SOAT was used to resolve (DL)-α-amino-ε-caprolactam, a lysine precursor, with the In In1979, 1979, SOAT SOAT was was used used to toresolve resolve (DL (DL)-α)--aminoα-amino-ε--caprolactam,ε-caprolactam, a lysinea lysine precursor precursor, with, with the the racemization being induced by potassium hydroxide in ethanol at 80 ◦C (Figure 13)[63]. Likewise, racemizationracemization being being induced induced by by potassium potassium hydroxide hydroxide in inethanol ethanol at at80 80°C °C (Figure (Figure 13 )13 [63]) [63]. Likewise,. Likewise, the enantiomers of the precursors of paclobutrazol, which served as an example for TCID, were also thet heenantiomers enantiomers of ofthe the precursors precursors of ofpaclobutrazol paclobutrazol, which, which served served as asan an example example for for TCID TCID, were, were also also obtained in high enantiomeric purity with SOAT by Black et al. in 1989 [64]. obtainobtaineded in inhigh high enantiomeric enantiomeric purity purity with with SOAT SOAT by by Black Black et alet. alin. in1989 1989 [64] [64]. .

Figure 13. (DL)-α-amino-ε-caprolactam can be resolved by SOAT to produce enantiopure lysine after FigureFigure 13. 13.(DL)- (DL)α-α-amino--aminoε--caprolactamε-caprolactam can can bebe resolved by by SOAT SOAT to to produce produce enantiopure enantiopure lysine lysine after hydrolysis. afterhydrolysis. hydrolysis.

LamellarLamellar epitaxies epitaxies between between crystals crystals of of opposite opposite handedness handedness could could be be a serious a serious problem problem for for PreferentialPreferentialPreferential CrystallizationC rystallizationCrystallization and and Kondepudi’sand Kondepudi’s Kondepudi’s experiments experiments experiments [19]. The [19] remedy[19]. The. The is to remedy rely remedy on deracemization, is is to to rely rely on on whichderacemizationderacemization has been, provenwhich, which has to has achieved been been prove prove 100%n ton e.e. to achieved achieved even when 100% 100%the e.e. e.e. crystals even even when of when the the two the crystals enantiomerscrystals of ofthe the two can two produceenantiomersenantiomers repeated can can produce epitaxies produce repeated [ 65repeated]. epitaxie epitaxies [65]s [65]. . Kondepudi’sKondepudiKondepudi’s experiments ’s experiments could could also also be becontrolled controlled by by the the addition addition of ofspecific specific specific chir chiral chiral impurities.al impurities. In In this this this process, process, process, to to a racemic to a racemic a racemic supersaturated supersaturated supersaturated solution solution solution of a conglomerate-forming of of a conglomerate a conglomerate-forming- system,forming system, a system, small amounta small a small ofamountamount R* additive of ofR* R*additive is addedadditive is to addedis the added medium. to tothe the medium. If R*medium. has aIf su R*Ifffi R*hascient has a degree sufficienta sufficient of analogydegree degree of with ofanalogy analogy R, the with nucleation with R, Rthe, the andnucleationnucleation growth and of and R growth and growth S become of ofR andR significantly and S become S become disignificantlyff significantlyerent and precipitationdifferent different and and leads precipitation precipitation to an excess leads leads of Sto in toan the an excess solid. excess Thisof ofS in eSff intheect the issolid. knownsolid. This This as effect the effect rule is knownis of known reversal as asthe [ 66the rule]. rule Of of course, ofreversal reversal there [66] [66] is. Of a.reversibility Of course course, there, there of theis ais rulereversibility a reversibility of reversal, of of whichthethe rule rule means of ofreversal thatreversal if, S*which, andwhich R*means means also crystallizethat that if S*if S*and as and aR* stable R*also also conglomerate,crystallize crystallize as asa R stable a could stable conglomerate, stereoselectively conglomerate, R could delayR could thestereoselectivelystereoselectively nucleation of delay R*, delay giving the the nucleation waynucleation to the of ofR* nucleation ,R* giving, giving way and way to growth tothe the nucleation nucleation of S*. It and is and also growth growth possible of ofS*. toS*. It induce isIt alsois also stereoselectivepossiblepossible to toinduce induce nucleation stereoselective stereoselective by using nucleation a polarizednucleation by laser by using using beam a polarized ina polarized a supersaturated laser laser beam beam solution. in ina supersaturateda supersaturated This process, knownsolution.solution. as ThisNon-Photochemical This process process, known, known Laser as asNon Induced Non-Photochemical-Photochemical Nucleation (NPLIN), Laser Laser Induced Inducedhas received Nucleation Nucleation some attention(NPLIN (NPLIN), [ 67) has, ]. has receivedreceivedIn the some case some attention of attention PE, the [67] final [67]. state. could also be controlled by an initial minor investment. As the initial step isIna Inthe total the case dissolutioncase of ofPE PE, the, ofthe final non-racemizable final state state could could also enantiomers also be becontrolled controlled (at least by by a inn a theinitialn initial context minor minor of investment. the investment. crystallization), As As the the theinitialinitial CSD step of step the is is solida total a total has dissolution no dissolution influence of on ofnon thenon-racemizable outcome-racemizable of the enant enant process.iomersiomers Successive (at (at least least applications in in the the context context of PE of lead of the the tocrystallization) the alternation, the of ( CSD), (+ of), ( the), (+ solid), etc., has a slightly no influence enriched on solid the outcome and the opposite of the process. series for Successive the liquid crystallization), the− CSD of− the solid has no influence on the outcome of the process. Successive phase,applications strongly of enrichedPE lead to in the (+), alternation ( ), (+), ( ), of (+ (),−), etc. (+), These (−), (+), alternations etc., a slightly are clearly enriched reproducible. solid and the applications of PE lead to the− alternation− of (−), (+), (−), (+), etc., a slightly enriched solid and the

SymmetrySymmetry 2020 2020, 12,, x12 FOR, x FOR PEER PEER REVIEW REVIEW 13 of13 18 of 18

oppositeSymmetryopposite 2020 seri , seri12es, xfores FOR forthe PEER the liquid REVIEWliquid phase phase , strongly, strongly enriched enriched in (+),in (+), (−), ( −(+),), (+), (−), ( −(+),), (+), etc. etc. These These alternation alternation13 of 18s s areare Symmetryclearly clearly reproducible.2020 reproducible., 12, x FOR PEER REVIEW 13 of 18 opposite series for the liquid phase, strongly enriched in (+), (−), (+), (−), (+), etc. These alternations 4.are Conclusions4.opposite clearly Conclusions reproducible. seri andes andfor Perspectives the Perspectives liquid phase , strongly enriched in (+), (−), (+), (−), (+), etc. These alternations Symmetryare clearly2020, 12 reproducible., 1796 13 of 17 4. ConclusionsCrystallizationCrystallization and Perspectives offers offers a variety a variety of ofmethods methods for for spontaneous spontaneous macroscopic macroscopic chiral chiral symmetry symmetry breaking.breaking. The The corresponding corresponding processes processes are are run run close close to equilibriumto equilibrium for forDeracemization Deracemization Induced Induced by by 4. CrystallizationConclusions and offers Perspectives a variety of methods for spontaneous macroscopic chiral symmetry 4.a Conclusions Fluxa Flux of of Energy Energy and Crossing Perspectives Crossing the the Suspension Suspension (DIFECS) (DIFECS) (Viedma (Viedma ripenin ripening, g TCID, TCID and and other other breaking. The corresponding processes are run close to equilibrium for Deracemization Induced by deracemizationderacemizationCrystallization variants variants offers where where a a varietysingle a single flux of flux methodsor severalor several for fluxes fluxes spontaneous of enof ergyenergy pass macroscopic pass through through chirala suspension) a suspension) symmetry a Flux of Energy Crossing the Suspension (DIFECS) (Viedma ripening, TCID and other or withorbreaking.Crystallization with a significant a Thesignificant corresponding odepartureff ersdeparture a variety fromprocesses from ofequilibrium methodsequilibrium are run forclose for spontaneousthe to the Kondepudi equilibrium Kondepudi macroscopic experiment for experiment Deracemization chiralwith with preferential symmetry preferentialInduced by breaking. deracemization variants where a single flux or several fluxes of energy pass through a suspension) Thesecondarysecondarya corresponding Flux nucleation of nucleation Energy, processes or , Crossing evenor even very are very thefar run farfrom Suspension closefrom equilibrium equilibrium to equilibrium (DIFECS) in thein thecase(Viedma for case of Deracemization Preferentialof Preferential ripening Enrichment, TCIDEnrichment Induced and (PE). by other(PE). a Flux of or with a significant departure from equilibrium for the Kondepudi experiment with preferential EnergyThoseThosederacemization processes Crossing processes include the variantsinclude Suspension an whereanimportant important a (DIFECS) single amplification amplification flux (Viedma or several of localof ripening, fluxeslocal fluctuations fluctuations of TCID energy andand pass and othertheir through their mechanisms deracemization mechanisms a suspension) are are variants secondary nucleation, or even very far from equilibrium in the case of Preferential Enrichment (PE). wherecurrentlycurrentlyor with a single the a the significantfocus flux focus of or manyof several departuremany studies fluxesstudies .from Interestingly, of. Interestingly, energyequilibrium pass as theas throughfor the systemthe system Kondepudi a departs suspension) departs more experiment more orfrom withfrom equilibrium with aequilibrium significant preferential, the, the departure Those processes include an important amplification of local fluctuations and their mechanisms are mechanicalmechanicalsecondary stress nucleation stress imposed imposed, or oneven onthe verythe system systemfar from has has toequilibrium beto besoftened softened in thein orderin case order ofto Preferential observeto observe the Enrichmentthe spontaneous spontaneous (PE). fromcurrently equilibrium the focus forof many the Kondepudi studies. Interestingly, experiment as withthe system preferential departs secondarymore from equilibrium nucleation,, orthe even very chiralchiralThose symmetry symmetryprocesses breaking breakinginclude (see an (see Tableimportant Table 1). 1Indeed,). amplification Indeed, in Viedmain Viedma of localripening ripening fluctuations performed performed and close their close to mechanisms equilibrium,to equilibrium, are farmechanical from equilibrium stress imposed in the on case the of system Preferential has to be Enrichment softened in (PE). order Those to observe processes the s includepontaneous an important all allcurrently sorts sorts of oftheabrasion abrasionfocuss of, breakagesmany, breakage studiess, defectss, . defectsInterestingly, induced induced as by the by shear system shear forces, departs forces, etc. more etc., are, are from beneficial beneficial equilibrium to to the , the amplificationchiral symmetry of breaking local fluctuations (see Table 1). and Indeed, their in mechanismsViedma ripening are performed currently close the focusto equilibrium, of many studies. advancementadvancementmechanical of stress chiral of chiral imposed symmetry symmetry on breaking. the breaking. system In theInhas the Kondepudi to Kondepudi be softened experiment, experiment, in order the to the mechanicalobserve mechanical the stress s pontaneousstress must must Interestingly,all sorts of abrasion as the systems, breakage departss, defects more from induced equilibrium, by shear the forces, mechanical etc., are stress beneficial imposed to on the the system be softerbechiral softer tosymmetry avoidto avoid primary breaking primary and (seeand secondary Tablesecondary 1). heteronucleation Indeed, heteronucleation in Viedma in ripeningthein the system. system. performed In theIn the case close case of toPreferentialof equilibrium,Preferential advancement of chiral symmetry breaking. In the Kondepudi experiment, the mechanical stress must hasEnrichmentEnrichmentall to be sorts softened (PE)of (PE)abrasion, the, in the very order verys ,large breakage tolarge departure observe departures, defects the from spontaneousfrom equilibrium induced equilibrium by chiralhas shear has to be symmetryto forces,associatedbe associated etc. breaking with, are with almost beneficial almost(see stagnant Table stagnant to1 the). Indeed, be softer to avoid primary and secondary heteronucleation in the system. In the case of Preferential incondi Viedmacondiadvancementtions.tions. ripeningIndeed, Indeed, of chiralfor performed forthe symmetry the latter latter case close casebreaking. a simple to a simple equilibrium, In magnetic the magnetic Kondepudi stirrer all stirrer sorts is experiment, sufficient ofis sufficient abrasions, to the returnto mechanical breakages,return the the solution stresssolution defects tomust to induced Enrichment (PE), the very large departure from equilibrium has to be associated with almost stagnant bynormal shearnormalbe softer condition forces, condition to avoid etc.,s of sprimary are crystallizationof crystallization beneficial and secondary to without the without advancement heteronucleation macroscopic macroscopic ofchiral chiral inchiral the symmetry symmetrysystem.symmetry In breaking breaking.the breaking case (see of (see InPreferentialTable the Table Kondepudi 1 1 conditions. Indeed, for the latter case a simple magnetic stirrer is sufficient to return the solution to experiment,below).below).Enrichment the (PE) mechanical, the very large stress departure must be from softer equilibrium to avoid has primary to be associated and secondary with almost heteronucleation stagnant in normal conditions of crystallization without macroscopic chiral symmetry breaking (see Table 1 thecondi system.tions. In Indeed, the case for of the Preferential latter case Enrichmenta simple magnetic (PE), stirrer the very is sufficient large departure to return from the solution equilibrium to has below).Table 1. Spontaneous macroscopic break of symmetry by means of crystallization. In theory, the final to benormal associatedTable condition 1. Spontaneous withs of almost crystallization macroscopic stagnant break without conditions. of symmetry macroscopic Indeed, by means chiral for of thecrystallization. symmetry latter case breaking In a theory, simple (seethe magnetic final Table 1 stirrer below).statestate of theof thesystem system is not is notpredictable. predictable. There There is a is stochastic a stochastic evolution evolution during during the thefirst first step step and and then then is sufficient to return the solution to normal conditions of crystallization without macroscopic chiral amplification.Tableamplification. 1. Spontaneous In DeracemizationIn Deracemization macroscopic Induced break Induced of bysymmetry bya Flux a Flux ofby Energyofmeans Energy ofCrossing crystallization. Crossing the theSuspension SuspensionIn theory, (DIFECS) the (DIFECS) final symmetryprocesses,stateprocesses,Table ofbreaking the 1. there Spontaneoussystem there are (see is areagonist not agonist Table macroscopicpredictable. effects1 effectsbelow). between break Therebetween ofisultra symmetrya ultrastochasticsound,sound, bmicrowaves,y evolution meansmicrowaves, of crystallization.during attrition, attrition, the TCID first TCID In stepand theory, and and damped dampedthethen final TCIDamplification.TCIDstate (and of(and photonsthe photons Insystem Deracemization for forislight not light-sensitive predictable.-sensitive Induced molecules). molecules). There by a isFlux a stochastic of Energy evolution Crossing during the Suspension the first step (DIFECS) and then Tableprocesses,amplification. 1. Spontaneous there are In agonist Deracemization macroscopic effects between Induced break ofultra by symmetry asound, Flux ofmicrowaves, byEnergy means Crossing ofattrition, crystallization. the TCID Suspension and In damped theory,(DIFECS) the final stateTCIDprocesses, of(and the photons system there forare is light agonist notpredictable.sensitive effects molecules). between There ultra is asound, stochastic microwaves, evolution attrition, during TCID the and first damped step and then TowardsTowards PreferentialPreferential HeterogenousHeterogenous Nearly Nearly- - 0: Stagnant0: Stagnant amplification.TCID (and photons In Deracemization for light-sensitive Induced molecules). by a Flux of Energy Crossing the Suspension (DIFECS) greatergreater EnrichmentEnrichment RacemicRacemic Crystals Crystals (HNRC) (HNRC) conditionsconditions to to Towardsprocesses, there arePreferential agonist effects betweenHeterogenous ultrasound, Nearly microwaves,- attrition,0: Stagnant TCID and damped TCID departuredeparture withwith possibility possibility of of staystay away away from from greater Enrichment Racemic Crystals (HNRC) conditions to (andfromTowardsfrom photons for light-sensitivePreferential molecules). alternatingalternatingHeterogenous homochiral homochiral Nearly - stable0: Stagnantstable departure with possibility of stay away from equilibriumTowardsequilibriumgreater Enrichment domains.domains.Racemic Largest CrystalsLargest domains domains(HNRC) equilibriumconditionsequilibrium to from alternatingHeterogenous homochiral stable greaterdeparture correspondingcorrespondingNearly-Racemicwith possibility to theto the minor Crystals ofminor stay away from 0: Stagnant departureequilibrium from from domains.alternating(HNRC)enantiomer Largest with homochiral domains possibility of equilibriumstable enantiomer conditions to stay equilibrium Preferential Enrichmentcorrespondingalternating to the homochiral minor Symmetryequilibrium 2020, 12, x FOR PEER REVIEW domains. Largest domains awayequilibrium from stable 14 of 18 domains. Largest domains correspondingenantiomer to the minor equilibrium ConglomerateConglomeratecorresponding forming forming to the Deracemization Conglomerateminorenantiomer enantiomer forming From soft to KondepudiKondepudi’s ’s systemsystem Moderate,Moderate, Induced by a Flux Conglomeratesystem forming medium for ExperimentExperiment ConglomerateFast racemization forming in Moderate,collisioncollision but collision but of Energy Crossing FastFast racemization racemizationsystem in in TCID Symmetry 2020, 12, x FORKondepudi PEER REVIEW’s system avoidModerate,but too avoid strong too 14 of 18 Kondepudi’sthe Suspension Experiment solutionsolutionsolutionConglomerate ororFast nonnonor racemization non--chiralchiral-chiral forming entityentity entity in Strongavoid toowith strong Experiment Fast racemization in collisionstrong shearing but Kondepudi’s solutionin solutionin systemsolution or non-chiral shearingshearingModerate, effects effects Deracemization(DIFECS): Conglomeratein solution forming avoidshearingFrom tooeff ectsstrongeffect soft to Experiment solutionFast orentity nonracemization-chiral in solution entity in collision but -InducedUltrasounds by a Flux A lamellar conglomeratesystem formedium attrition for Towards Conglomeratein solution forming shearingavoid tooeffects strong Deracemization-Microwaves Induced bysolutiondoes notor non hinder-chiral entity enhancedFrom soft i.e. to greater of Energy Crossing Fast racemizationsystem in TCID a Flux of Energy Crossing in solution mediumshearing for effects TCID the-TCID Suspension solutionderacemizationFast or racemization non-chiral entity in ViStrongedma with intensity of the Suspension (DIFECS): Strong with -Viedma Ripening solution or non-chiral Ripening theTowards -Ultrasounds(DIFECS): in solution shearingshearing eff ecteffect for entity in solution greater (close--MicrowavesUltrasounds to 0) A lamellar conglomerate attritionfor attrition enhanced mechanicalTowards A lamellar conglomerate intensity of the -TCID i.e., Viedma stressgreater -Microwaves doesdoes not not hinder hinder enhanced i.e. mechanical -Viedma Ripening (close to 0) Ripening -TCID deracemizationderacemization Viedma intensitystress of -Viedma Ripening Ripening the (close to 0) mechanical It is also possible to control a macroscopic chiral symmetry breaking such as Viedmastress ripening, TCID and its variants, or SOAT or preferential crystallization. Some of these elegant processes could be made productive enough for industrial applications. Thus, there are fundamental and applied It is also possible to control a macroscopic chiral symmetry breaking such as Viedma ripening, interestsTCID and in its these variants crystallization, or SOAT or preferential processes associated crystallization. with Some macroscopic of these elegant symmetry processes breaking. could

be made productive enough for industrial applications. Thus, there are fundamental and applied Authorinterests Contributions: in these crystallizationBoth authors processes contributed associated to this with publication. macroscopic All symmetry authors havebreaking. read and agreed to the published version of the manuscript. It is also possible to control a macroscopic chiral symmetry breaking such as Viedma ripening, Funding:AuthorTCID Contributions: andThis its review variants isBoth part, or authors SOAT of a PhD contributed or preferential funded to by thi the crystallization.s publication. University All of Some Rouenauthors of Normandy. thesehave readelegant and processes agreed to couldthe published version of the manuscript. Conﬂictsbe made of Interest:productiveThe enough authors for declare industrial no conﬂict application of interest.s. Thus, there are fundamental and applied Funding:interests This in review these iscrystallization part of a PhD fundedprocesses by theassociated University with of Rouen macroscopic Normandy. symmetry breaking. Conflicts of Interest: The authors declare no conflict of interest. Author Contributions: Both authors contributed to this publication. All authors have read and agreed to the published version of the manuscript. List of Symbols and Abbreviations Funding: This review is part of a PhD funded by the University of Rouen Normandy. CSD: Crystal Size Distribution DIFECS:Conflicts ofDeracemization Interest: The authors Induced declare by ano Flux conflict of Energy of interest. Crossing the Suspension e.e.: Enantiomeric excess = (R − S)/(R + S) GRD:List of SymbolsGrowth and Rate Abbreviations Dispersion HNRC:CSD: HeterogeneousCrystal Size DistributionNearly-Racemic Crystals NPLIN:DIFECS: Non Deracemization-Photochemical Induced Laser Induced by a Flux Nucleation of Energy Crossing the Suspension PC:e.e.: PreferentialEnantiomeric Crystallization excess = (R − S)/(R + S) PE:GRD: PreferentialGrowth Rate Enrichment Dispersion SOAT:HNRC: SecondHeterogeneous Order Asymmetric Nearly-Racemic Transformation Crystals TCID:NPLIN: TemperatureNon-Photochemical Cycle-Induced Laser DeracemizationInduced Nucleation US:PC: UltrasoundPreferential Crystallization PE: Preferential Enrichment ReferencesSOAT: Second Order Asymmetric Transformation 1. TCID:Coquerel , G.Temperature Review on the Cycle Heterogeneous-Induced Deracemization Equilibria between Condensed Phases in Binary Systems of US:Enantiomers. Ultrasound Enantiomer 2000, 5, 481–498. 2. Clevers, S.; Coquerel, G. Kryptoracemic Compound Hunting and Frequency in the Cambridge Structural ReferencesDatabase. CrystEngComm 2020, doi:10.1039/D0CE00303D. 3. 1. Fábián,Coquerel L.; ,Brock, G. Review C.P. on A the List Heterogeneous of Organic Kryptoracemates.Equilibria between Acta Condensed Crystallogr. Phases B 2010in Binary, 66, Systems 94–103, of doi:10.1107/S0108768109053610.Enantiomers. Enantiomer 2000 , 5, 481–498. 4. 2. Tiekink,Clevers, E.R.T. S.; Coquerel, Kryptoracemates. G. Kryptoracemic In Advances Compound in Organic Hunting Crystal and Chemistry Frequency, Comprehensive in the Cambridge Reviews Structural 2020; Springer:Database. Cham, CrystEngComm Switzerland, 2020 2020;, doi:10.1039/D0CE00303D. Chapter 19, pp. 381–404. 5. 3. Rekis,Fábián, T. Crystallization L.; Brock, C.P. of Chiral A List Molecular of Organic Compounds: Kryptoracemates. What Can Acta Be LearnedCrystallogr. from B the2010 Cambridge, 66, 94–103 , Structuraldoi:10.1107/S0108768109053610. Database? Acta Crystallogr. Sect. B Struct. Sci. Cryst. Eng. Mater. 2020, 76, 307–315, 4. doi:10.1107/S2052520620003601.Tiekink, E.R.T. Kryptoracemates. In Advances in Organic Crystal Chemistry, Comprehensive Reviews 2020; Springer: Cham, Switzerland, 2020; Chapter 19, pp. 381–404. 5. Rekis, T. Crystallization of Chiral Molecular Compounds: What Can Be Learned from the Cambridge Structural Database? Acta Crystallogr. Sect. B Struct. Sci. Cryst. Eng. Mater. 2020, 76, 307–315, doi:10.1107/S2052520620003601.

Symmetry 2020, 12, 1796 14 of 17

List of Symbols and Abbreviations

CSD: Crystal Size Distribution DIFECS: Deracemization Induced by a Flux of Energy Crossing the Suspension e.e.: Enantiomeric excess = (R S)/(R + S) − GRD: Growth Rate Dispersion HNRC: Heterogeneous Nearly-Racemic Crystals NPLIN: Non-Photochemical Laser Induced Nucleation PC: Preferential Crystallization PE: Preferential Enrichment SOAT: Second Order Asymmetric Transformation TCID: Temperature Cycle-Induced Deracemization US: Ultrasound

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