Mechanical chiral resolution FINAL(with modifications of

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Pleasedonotadjustmargins


a. UniversitédeStrasbourg,CNRS,ISIS,8alléeGaspardMonge,67000Strasbourg,[email protected]

b. DepartmentofPhysics,UniversityofStrathclyde,GlasgowG4ONG,UnitedKindom

c. SchoolofPhysicsandAstronomy,UniversityofGlasgow,GlasgowG128QQ,UnitedKingdom

d. Nordita,RoyalInstituteofTechnologyandStockholmUniversity-Roslagstullsbacken23,SE-10691Stockholm,Sweden

Received00thJanuary20xx,Accepted00thJanuary20xx

DOI:10.1039/x0xx00000x

MechanicalchiralresolutionVincentMarichez,aAlessandraTassoni,aRobertP.Cameron,bStephenM.Barnett,cRalfEichhorn,dCyriaqueGenet,aandThomasM.Hermans*,a

Mechanicalinteractionsofchiralobjectswiththeirenvironmentarewell-establishedatthemacroscale,likeapropelleronaplaneorarudderonaboat.Atthecolloidalscaleandsmaller,however,suchinteractionsareoftennotconsideredordeemed irrelevant due to Brownian motion. As we will show in this tutorial review, mechanical interactions do havesignificant effects on chiral objects at all scales, and can be induced using shearing surfaces, collisions with walls orrepetitivemicrostructures,fluidflows,orbyapplyingelectricaloropticalforces.Achievingchiralresolutionbymechanicalmeans is very promising in the field of soft matter and to industry, but has not received much attention so far.

1IntroductionThe first reference to chirality traces back to 360 BC and isattributedtoPlatoinhisdiscussionwithpeersaboutthevisionmechanism.1 More than two millennia later this notion wasrevisited by young Louis Pasteur who identified "handed-molecules",2basedonpreviousworkbyFresnel3oncircularlypolarized light.Molecules that could not be superposedwiththeirrespectivemirrorimageswerefoundtohavetheuniqueabilitytorotatetheplaneofpolarizedlightandwerethussaidto display optical activity, at that time denoted as“dissymmetry”.Pasteur’sattributionofthelattereffecttothethree-dimensional arrangement of atoms in molecules waslater confirmedby LeBel4 and independentlyby van ’tHoff5.However, since dissymmetric objects were not necessarilyasymmetric, the word “dissymmetry” (although still widelyused in France) was slowly replaced by “chirality” as putforwardbyLordKelvin6.Lifeischiral,andmostbiologicalprocessesfoundinnaturearestereospecific, i.e. sensitive to the handedness of themolecules involved. As a result, most drugs must beadministrated as a single enantiomer (eutomer), since itsenantiomeric formcanhavedetrimental effects (distomer).7,8Chiral resolution is therefore a cornerstone of mostpharmaceuticalprocessesandiscurrentlydominatedbychiralcolumn chromatography. In recent years, efforts have beenmade to find alternatives to column chromatography, andmorespecifically to theuseof itschiral stationaryphase.Thelatterconsistsofchiralmoleculesimmobilizedonmicron-sized

beads, which are packed inside a cylinder to form a chiralcolumn. A mixture is separated, because one of theenantiomers interacts more strongly with the chiral column.Drawbacks of column chromatograph in general are therelatively high pressures needed to pump through thestationary phase, and a large consumption of solvent.Furthermore,chiralcolumnsareexpensive.Last year, two interesting new methods of chiral separationhavebeenreported,oneinvolvingtheinteractionwithachiralperpendicularly magnetized substrates9, and a second basedon a homochiral enantiomer sieving membrane10, which arenot based on mechanical resolution and thus beyond thescope of this tutorial review. Here, we focus on chiralresolutionmethods from the colloidal to the singlemoleculelevel based on mechanical interactions. Some of thesemethodsmayonedaybecontenderstoprovideanalternativeforcurrentresolutionmethods,andparticularlychiralcolumnchromatography.First, we want to address the scepticism that surroundsmechanicalresolution.ChiralresolutionusingshearflowswasinitiallysuggestedbyHowardetal.11,butdidnothaveawidefollow-up. Later, Tencer and Bielski put forward predictionsthat chiral resolution using shear flowwill neverwork at themolecular scale.12 Indeed, smaller object experience moreviolent thermal fluctuations, and since chiral lift depends ontheorientationofthechiralobject(aswewillseelater),thisisproblematic. Specifically, rotational diffusion will constantlyreorient the chiral object in random directions. Let usintroducetherotationaldiffusioncoefficientDrforasphericalobjectofdiameterd(eq.1):

3 )/ (r BD k T dph= (1)

wherekBistheBoltzmannconstant,ηthedynamicviscosityofthe liquidmedia,andT theabsolute temperature.TheDebyerelaxationtimeτhappenstobeagoodorientationpersistenceindicator(eq.2):

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)1 / (2 rDt = (2)

A 1 micrometre sized object in aqueous solution at roomtemperature can thus maintain its orientation for a fewseconds whereas a chiral molecule re-orients already aftersome nanoseconds. Rotational diffusion is thus a seriouslimitation to chiralmigration. The second limitation is due totranslational diffusion (i.e., a randomwalk), also arising fromthermal fluctuations that tends to counterbalance changes inconcentration built up by mechanical resolution. Consideringan approximately spherical object, one can define thecompetition of chiral drift versus a random walk bynumerically deriving the merit number /y yb = . Here, yrepresentsthedistancebetweentwoenantiomersafteratimetdriftingawayfromeachotherbyanangle𝛼 (seeFig.1)andcanbederivedfromtheStokesequationasfollows(eq.3):

/ (3 )y F t da ph@ (3)

where F is the drag force, η is the dynamic viscosity of themedium,dthediameterofthesphericalobjectandttime.Asmentioned before, chiral migration has to compete withrandomwalkalongthey-direction,whichaccordingtoEinsteinandSmoluchowski,13,14canbequantifiedby(eq.4):

2y Dt= (4)

with D being the translational diffusion coefficient. ThreedistinctregimescanbedistinguishedinFig.1c:

• 𝛽>>1:Enantiomericresolution.• 𝛽≃1:Enantiomericenrichment.• 𝛽<<1:Noresolution.

The smaller the object, themore randomwalk outcompeteschiraldrift,andatthenanometrescale(cf.bluebottomlineinFig. 1), chiral resolution nor enrichment (i.e., not fullresolution)canbeachievedevenaftermanymillennia!

Fig.1Competitionbetweenchiraldriftandrandomwalk.a)Illustrationofthedistancebetween 2 enantiomers after an opposite chiral drift by𝛼/2 during a time t. Chiralmechanicalresolutionislimitedbythecompetitionwithb)randomwalkresultingfromthermalfluctuations.c)Meritnumberplottedversustimeforthreesizes(1mm,1µm,and1nm)oforientedchiralobjectsassuming𝛥𝜌=1650kg/m3and𝛼=0.01.AdaptedfromRef.12withpermissionfromJohnWiley&Sons.

Thistheoreticalpredictionprovidedableakprospectforactualexperimental studiesonmechanical chiral resolution. In spiteof the latter predications, several experimental studies haverevealed promising chiral separations. From these successful

experiments it becomes clear that the above picture is toosimplistic and thatmoreaccurate theoreticaldescriptionsareneededinthefuture.

2CrystalsortingIn 1880, the Curie brothers reported what was to be calledpiezoelectricity. In their work, they were already convincedthat this particular effect was dependent on the crystalstructure of the material considered. This early discoveryaroused the curiosity of Pierre Curie who intensivelyquestioned himself about the role of symmetry on physicalphenomena:“Whencertaincausesproducecertaineffects,thesymmetryelementsofthecausesmustbefoundintheeffectsproduced. When certain effects reveal a certain asymmetry,thisasymmetrymustbefoundinthecauseswhichgivebirthtothem.”15Thisstatementisnowknownasthe"CuriePrinciple"andcauseditsinventortothinkaboutthepossibilitytocreatean asymmetric environment (useful to perform chiralseparation) by juxtaposition of symmetric forces, such as anelectric field with a magnetic field. This vision was howevervigorously criticized by Barron who suspected it to provide“false chirality”.16 Nevertheless, the possibility to resolveenantiomerswithouttheuseofanyenantioenrichedreagent,material or force was found to be really intriguing and wasnotably discussed by Welch17 and De Gennes18. The latterhypothesized that a chiral crystal floating on a liquid surfaceand pushed by a uniform force (such as a jet of air) wouldexperience friction forcewhosedirectiondirectlydependsontheshapeofthesurfaceincontactwiththeliquidsurface(seeFig.2).

Fig.2AChiralcrystalfloatingonaliquid(thecrystalisassumedtobemuchlessdensethantheliquidsothatitsfacebarelytouchestheliquid).Whenhavingacloserlookatits contact area (red, hatched, top view), the velocityv is not parallel to the driftingforceF.Infact,wecanassumealinearrelationbetweenthevelocityvandthereaction

force: R vV- = with V beinga2x2symmetricmatrix.Theanisotropyof V arisesfromthe shape anisotropy of the face in contact with the liquid. If we consider itsenantiomer (inblue), the surfaceanisotropywould yield anoppositemotion. If bothcrystalsweredroppedattheliquidsurfaceandblownbyauniformairjet,theywouldtravel in opposite direction and thus spatially separate. Adapted from Ref. 18 withpermissionfromEurophysicsLetters.

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In short, enantiomorphic crystals could theoreticallyexperienceanoppositetangentialmotioninsofarastheirfacesincontactwiththeliquidaremirrorimagesofeachother.Thissuggested method shows that a chiral drift can arise fromsymmetricforces(heretheairjet).Forthistobepossible,thechiral object must however experience some sort of motionwithrespecttothefluidjustlikeaboatsailingontheseathatispushedbythewindandwhosemotionisconstrainedbythewaterline. In other words, to perform mechanical chiralresolutionwithouttheuseofanenantioenrichedreagent,wecanlistrequirementstofulfill:

1. The top, the bottom, the front and the back of thechiral object must be spatially differentiated (i.e.:orientationpersistence).

2. The chiral object must be submitted to some sort offorce able to move it along (by external force orpassiveadvection).

Originallypuzzledby the fact thatDutchbeacheswere foundtobepopulatedbyunequalquantitiesofleftandrightshellsofbivalves19, Howard11 was the first to experimentally exploremechanical chiral separation. In fact, he imagined a way toautomate the manual crystal sorting performed by LouisPasteur2, with the use of fluid flows. To this end, he used arotating drum containing millimetre sized chiral crystalsimmersed in a poor solvent (i.e., to prevent dissolution). Thedrumrotatesaboutitsaxisandtherotationalspeedischosento keep the particlewell up the side of the drumbut not sohigh that it rotateswith the fluid. By doing so, orientation ismaintainedbygravitationalforcesproducedbynearcontactofonecrystal facewiththedrumwallandbythefinitedistancebetweenthecentreofgravityoftheparticleandthecentreofhydrodynamicreaction(wheretorqueisapplied).Thanks to this orientation persistence and gravity-assistedmotion,Howard11couldeasilyresolveenantiomorphiccrystalsthatwouldmovetowardsoppositeendsofthedrum(seeFig.3).

Fig. 3 Method used by Howard11 to automate Pasteur’s manual crystal sortingprocedure2. It consists of a rotating drum containing millimetre sized chiral crystalsimmersedapoor solvent.A fine interplaybetween the centreofmassof the crystaland the reactor force centre facilitates orientation persistence. The weight of thecrystalforcesitto"fall"alongthewallofthedrum.Eachenantiomermigratestowards

aspecificendofthedrum.AdaptedfromRef.11withpermissionfromJohnWiley&Sons.

Thismethodcouldeasilyresolveenantiomorphiccrystalseventhoughforsomecrystals,thetop/downspatialdifferentiationisdifficulttomaintain.Thatis,somecrystalscanhavedifferentfaceswithnear equal probability tobe facing thewall of thedrum.This incompleteorientation can lead to reversalof themigration direction thus preventing enantiomers to beresolved. Similar behaviour waswitnessed for large amountsofcrystalswhichcan"disturb"eachotheraswellasconstantlyreorient. Thequestion arises, therefore,whether such a typeofmechanicalresolutioncanbescaleddowntothemolecularlevel. Welch17 imagined an elegant new way to achieve thelatter. He proposed to use electrophoresis to induce adirectional motion of chiral charged amphiphilic moleculestrapped at an interface between an organic and aqueousphase(seeFig.4).

Fig.4ElectrophoresisdeviceimaginedbyWelch17toseparatemolecularenantiomers.The chamber is filled with immiscible solvents to form a liquid-liquid interface. Anelectricfieldisappliedalongthechamber.Themoleculesneedtobeamphiphilictobetrappedattheinterface(top/downspatialdifferentiation),andchargedtobeorientedand pulled by one electrode. We show an example of a positively charged Binolamphiphilic derivative that fulfills the above requirements. If introduced through theanode,bothmoleculesshouldmigrate inoppositedirectionsatthe interfacetowardsthe cathode where they could be collected separately. Adapted from Ref 17 withpermissionfromTaylor&FrancisGroup.

The electric fieldwas expected to both inducemotionof themolecules and to orient the charged part of the moleculestowards one electrode leading to a front/back spatialdifferentiation.Trappingthemoleculesbetweenthetwoliquidphases was expected to orient the molecules (whosehydrophilic part would stay in the aqueous phase while itshydrophobic one would remain in the organic phase). Theinterface would thus allow a top/bottom spatialdifferentiation. All requirements being fulfilled, theenantiomers should then "sail" in this interface, migrating inopposite directions (in the xy plane). Despite its greatelegance, thismethodhasyet tobeexploredexperimentally.Moreover,theorientationofthemoleculefacingtowardstheanode will be rather inefficient due to thermal fluctuations,and the approach would be applicable only to very specificchargedamphiphilicmolecules.

3Resolutioninarheometer





We previously discussed about different mechanical chiralresolution methods that required the chiral objects to bemoved by forces. However, when considering the molecularscale, the list of available methods to actuate the chiralmolecules is reduceddrastically.Electrophoresis,asdiscussedabove, would only work with charged molecules. Similarly,magnetophoresiswouldrequirestronglyparamagneticspeciesandveryhighmagneticfieldgradients.It isnotstrictlynecessary toexertadirect forceonthechiralobject (thusachievingdriven transport).Onecanalso relyon(passive transportusing) fluid flows toproduce chiral specificdriftsprovidedthatsymmetryisbrokenbeforehand.

Fig. 5 Aqualitativeunderstandingof chiral lift on a twisted ribbon.Blue and yellowshadingindicatethetopandbottomhalvesoftheribbon.Thedragforceactingonthetoppartoftheribbon(cf.orangearrow)hasacomponentalong+zwhereasthedragforce acting on the bottom part of the ribbon is along the –z direction. a) In asymmetric flow profile, both sides are experiencing equally large opposing forces,whichcanceleachotherout.b)Inanasymmetricflow,thenetdragforceactingontheribbon is ≠ 0 and chiral drift is possible (note: compare the lengths of the green vs.whitearrows).

Letusconsideraright-handedtwistedribbon.Thisribboncanbeseparatedintwohalvesalongitslongitudinalaxis(cf.Fig.5in blue and yellow).We assume for simplicity that the chiralobjectisalignedintheflowdirection.Whentheribbonmovesthrough the fluid, it experiences adrag forceopposite to thedirection of the oncoming flow (where the velocity isproportional to the black arrows in Fig 5, while drag force isrepresented by orange arrows). The interaction of the fluidwiththebluesideofthehelixwillgenerateadragforcewithacomponent along the +z direction (white arrow), while theinteractionwiththeyellowhalvegeneratesadragforceinthe

–zdirection(greenarrow). Inauniformvelocityprofile,bothsidesofthehelixexperiencetheexactopposite lift forces:nochiral drift would be observed (Fig 5a). However, if the flowprofileisasymmetric(seeFig.5b),theforcesononehalvewilloutcompetetheotherhalf(hereyellowsurpassesblue)andanoverallchiraldriftwillemerge.Inotherwords, it isnotnecessary tomoveanorientedchiralobject in fluids to create a chiral-specific lift: shear flows cancarrytheparticlesalong instead.Usingshearflowsallowstheexperimentalisttofocusonlyontheorientationpersistenceofthechiralobjectandnolongeronthewaystomoveitalong.

3.1Taylor-Couetteflow

In1890,MauriceCouettepubliclypresentedanewconcentricviscometer, which he had used to accurately measure theviscosityoffluidsandlaterontostudyboundaryconditionstomakesurethatthe“no-slip”conditionwasverifiedforarangeofmaterialsandfluids.20Hehadnoideaatthattimethatmorethan a century later his rheometer would have had anothervirtue:thecapabilitytodiscriminatebetweenenantiomers.Letus consider a Couette cell made of two concentric cylinderswithRobeingtheouterradiusoftheinneroneandRitheinnerradius of the outer one. The outer cylinder turns with anangularvelocityΩ.Theshearrateataradialpositionrcanbedefinedas(eq.5):

!γ = 2Ω / (r 2 (1 / Ro2 − 1 / Ri

2 )) (5)

Theanisotropyoftheflowgeneratedintheannulargapallowsdiscriminationbetweenenantiomers(asexplainedinFig.5).Inotherwords, iftheoutercylinder isrotatedclockwise(CW),aleft- handed helix (blue) will experience a chiral lift in thevorticity direction and will migrate to the top of the cell. Aright-handedhelix(red)willexperienceanoppositeforceandwillthusmigratetowardsthebottom.Iftherotationdirectionisreversed,themigrationdirectionsarealsoreversed(seeFig.6).In a compelling numerical simulation, following Kim’s initialwork21, Makino and Doi22 studied the migration of twistedribbons in (circular) Couette flow. They showed thatenantiomers move in opposite directions with a velocityzv expressedbyeq.6:

lim( ( )) /z tv Z t t

®¥= (6)

where Z(t) is the average z–coordinate of the 1000 particlesconsidered in their simulation. Theywere able to verify howfar this mechanical chiral lift force could be scaled down intermsofparticle size. Inparticular, theydefined two regimesdependingontheparticlesize(eq.7andeq.8):

zv agµ when / 1rPe Dg= ³ (7)

and

2( / )z rv a Dg g= when / 1Pe Drg= £ (8)

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with a being the width of the twisted ribbon and Dr therotational diffusion coefficient. For a given shear rate,decreasing the size of the chiral object will thus increase itsrotationaldiffusionresultinginthelossofthechiralliftforceitexperiences.Tocounterthiseffectandrecoveradecentchirallift, one can either increase the shear rate (by increasing theradius of the cylinders, decreasing the gap, or increasing therotation speed) or try to decrease the rotational diffusion ofthe chiral objects by aligning them with an electric field forexample. According to their numerical simulation, a set ofconcentriccylinderswitha10cmradius,andagapdistanceof1mmrotatingat100rpmcouldbeenoughtocausemigrationofa10nmchiralobjectover1mminafewminutes.

Fig.6ChiraldriftexperiencedbytwoenantiomericstructuresinaTaylor-Couettecell.At time t0 a racemic mixture is placed in the cell. After a certain time t full chiralresolutionisachieveddependingontheshearrateandthesizeofthehelices.

Hermansetal.23 implementedanexperimentalmodel systemto separate enantiomers using a circular Couette flow.Hermans found thatby confining2D chiral objects at theair-liquid interface in the Couette gap, an in-plane chiral forcecouldbeinduced(i.e., intheradialdirection).23Thisapproachallowed them to providenot only an averagedriftingmotionbutalsoadetailedrotationalandtranslationalmotionintime,whichcould inturnbecomparedtothehydrodynamicmodeltheydeveloped.SU8photoresistchiralobjectswerefabricatedand placed at the surface of silicon oil in between the twocylinders. The particles revolved around the cell with aconstant velocity v𝛳 but experienced large variations in theirangular velocity ω known as the so-called Jeffery orbit for

elongatedparticles.ThecircularCouetteflowinducedachiralspecificradialdriftasshowninFig.7.The directionality and amplitude of the observed effect,according to their numerical considerations, are expected tobe related to the shape of the chiral object, the largest driftvelocitiesbeingexpectedforasymmetricparticleswithahighaspectratio.Aseriesofstochasticsimulationsallowedthemtoassess the possibility to discriminate between smallerenantiomers (micro- or nanometre scaled chiral objects).Interestingly,thein-planechiraldriftscaleslinearlywithtime,whereas diffusion scales with the square root of time. Thelatter suggests that over long times, chiral drift couldoutcompete diffusion at the molecular scale. A three-dimensional version of these experiments is currently beingstudiedbyHermansandco-workers.

Fig. 7 Chiral resolution at the air/liquid interface using a Taylor-Couette cell. a) TopviewoftheCouette-cellused,consistingoftwoconcentriccylinders.TheoutercylinderhasaradiusRowhiletheinneronehasaradiusRi.ThegapG=Ro–Riisfilledwithoil.Right-(blue)andleft-handed(red)chiralobjectsarefloatingattheair-liquidinterface.Once theoutercylinderstarts turningwithanangularvelocity𝜔cell,b)onecanseeachiral specific drift in the radial position (!" = (!"-&')/*) of the enantiomer R or S,which also depends on the turning direction, clockwise (CW) or counter-clockwise(CCW).AdaptedfromRef.23withpermissionfromSpringerNature.

3.2Rotatingparalleldiscs

A different type of rheometer consists of two parallel discs,spaced by a distance h (see Fig. 8). The upper disc rotateswhereastheloweroneisfixed.MakinoandDoi24showedbothexperimentally and numerically that shear flow in such aparallel-plate rheometer (see Fig. 8 ) could resolveenantiomers at the millimetre scale. Specifically, theyfabricatedchiralcubesbycuttingtwoedgestoproduceD-and





L-particles. The two cuboid enantiomers exhibited a netopposite drift in the vorticity direction. One enantiomer willexperienceadrift towardsthecentreof thediscwhereastheotherwillmigrate towards the outer radial region (see Fig. 8c). For theopposite rotationdirection, for symmetry reasons,themigrationdirectionsofeachchiralobjectalsoreversed.Asmentioned before,22 the migration velocity <vz> depends onthePecletnumber(Pe)andisgivenas(eq.9andeq.10):

2zv Ba Peg= when / 1rPe Dg= £ (9)

and

zv Cag= when / 1rPe Dg= ³ (10)

whereBandCarechiralityindexconstants(iftheobjectisnotchiral,BandCareequaltozeroandthereisnodrift),aistheparticlesizeand𝛾istheshearrate.

Fig.8Rotatingparalleldiscssetup24.a)Bothdiscsareparallel.Theupperonecanturnwhiletheloweronecannot.Thevelocityprofilegeneratedisdepictedinblue.b)D-andL-particlessculptedinsiliconrubber.c)OncetheD-andL-particlesareimmersedinanalmostdensitymatchedsiliconoilinthegapandwhentheupperdiscisrotatingCCW(inthisparticularcase),bothenantiomersmigrateoppositelyinthevorticitydirection(along the raxis).Adapted fromRef.24withpermission fromthePhysicalSocietyofJapan.

Once again, one can notice that Makino and Doi24 wereconcernedaboutthepossibilitytodownscalethiseffectdownto themolecular level. In theirmodel, the drift velocity doesnot scale linearly with the shear rate, which forces one todrastically increase the shear rate to be able to resolvemolecular enantiomers. Their experimental results werenumerically verified and they could correctly predict thedirection of migration of the chiral objects, but interestinglythe observed drift velocity ismuch higher than predicted (bytwoordersofmagnitude).Theyattributedthisdiscrepancytotheeffectof thewall thatcouldbetteralign theparticle,andthusenhancetheeffectoftheflow.

4ThepropellereffectIn 1978, Baranova and Zel’dovich25 predicted that the“propellereffect”couldbeusedtoseparateenantiomers inaracemic mixture. More specifically, the use of a radio-frequency electric field of rotating polarization can couple totheelectricdipolemomentofamoleculeinducingitsrotation.A subsequent rotational-to-translational coupling then allowsenantiomers to migrate in opposite directions (due to theirinherent opposite right-left asymmetry). This effect wasexpected to survive the randomization of Brownian motionandyet,noexperimentalevidencewasprovideduntilrecently.

4.1Atthecolloidalscale

In2013,Fisherandco-workers26performedaproofconceptatthe colloidal scale (Fig. 9). The latter was considered as anideal scale to study the propeller effect where the nano-structures used could be seen as “colloidal molecules” inmesoscopic systems. In this way, it was possible to directlyvisualizethephenomenon,whichwouldbeverychallengingatthemolecular scale. In their study, they replaced the electricdipolarmoleculeandtherotatingelectricfieldwithamagneticfield acting on a magnetic dipolar colloid. The colloidalstructures used were grown via Glancing Angle Deposition(GLAD), which allowed the production of large quantities ofhelices of defined size and shape. The colloidal helices aremagnetized orthogonally to their long axis. By doing so, theyacquire a magnetic moment that allows a magnetic field toapplyatorque𝜏magonthehelix(eq.11):

mag m Bt = ´ (11)

Wherem is the magnetization of a colloidal helix and B themagneticfield.Themicrometreobjectsarealsosubjectedtoaviscousdrag𝜏drag(eq.12):

3drag XRt h= W (12)

where R is the colloid’s characteristic size, X is a scale-freegeometry factor that depends on the shape, η is the fluidviscosity,andΩ is its speedof rotation.At lowmagnetic fieldrotation velocities, themomentof theparticlewill alignwiththefieldandmaintainaconstantvelocityΩ=ωwiththetwotorques balancing each other. The subsequent rotational-to-translational coupling allows the structures to move insolution.Forsymmetryreasons,twoenantiomerswillmigrateinoppositedirectionwithavelocity𝜐(eq.13):

u e= W (13)

where 𝜀 is the rotational-to-translational coupling constant,which directly depends on the pitch of the helix thussuggesting that the "degree of chirality" dictates thepropulsionefficiency.Theauthorscomparedthestrengthofthedipolecouplingwiththermalfluctuations(eq.14):

/mag BS mB k T= (14)

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Smag is of the order of 103, which means that in their

experiments the dipole coupling ismuch larger than thermalfluctuations, thus explaining the separation observed.However,ifwegobacktoBaranova’sidea(electricfielddrivenmolecules)25, the structures are much smaller, and thepropulsion efficiency is on the order of 10–4. Based on thelatterobservations,resolutionofenantiomersatthemolecularscaleseemshopeless.

Fig.9Thepropellereffectat thecolloidalscale.a)SEMimagesofawafercontaininghelices representative of those used in Ref. 26 obtained through glancing angledeposition.Scalebaris2𝜇m.b)Whenafieldisapplied,themagneticmomentsofthehelices(perpendiculartotheirlongaxis)alignswiththefielddirectionandrotateswithit.Therotational-to-translationalcouplingthenallowsright-andleft-handedhelicestomove in opposite directions. c) Relative intensity changewith time of the scatteringintensityacrossthewidthofa5mmcuvettefilledwithasuspensionofright-or left-handedcolloidsinarotatingmagneticfieldofastrengthof50Gaussandafrequencyof40Hz.AdaptedfromRef.26withpermissionfromtheAmericanChemicalSociety.

4.2Atthemolecularscale

Clemensandcoworkers27neverthelessdidattempttoperformchiral separation at the molecular scale. To this end, theysurroundedacapillarytubingwithfourelectrodes(seeFig.10)alternativelyswitchingbetween“highfield”(1100V)and“zerofield”valuesallowingaproper rotationof theelectric field infour phases. Each phase corresponds to a 90° turn, and wasabletoorientachiralbinaphthylmolecule.Asexplained,enantiomerswereexpectedtomoveinoppositedirections with a translational velocity u resulting from therotational-to-translationalcouplingandexpressedby(eq.15):

( ) ( ) ( )ref eff corru L u E A E F E= (15)

whereueff(E)istheeffectivemolecularrotationfrequency,Lrefis the valueof onedisplacementperone revolutionandAcorr

(E) is the angular correction factor and accounts for therandom orientation of the molecular propeller’s axis. F(E)corresponds to the “responding” fraction of molecules thatfollow the changes in electric field direction resulting indirectional propeller motion. In fact, a large fraction ofmolecules rotates equally in both directions and thereforetheydonotcontributetopropellerpropulsion.

Fig. 10 Rotating electric field-assisted propeller effect. a) Cartoon representing theseparationcapillarychambersurroundedbyfourelectrodes(A,B,CandD).b)Cross-sectionalcartoonsshowinghowtheelectric fieldrotates.Onecandecomposea360°completerotationinfourphasescharacterizedbyfourconsecutivetimepoints(t1,t2,t3and t4). c) Voltage of the four electrodes as a function of time during one cycle. d)Predicted direction of migration of R- and S- forms of 1,1’-Bi-2-naphtholbis(trifluoromethanesulfonate) when submitted to a clockwise rotating electric field.Thefieldrotatesaboutthex-axisintheyzplaneandformsanangleαwiththedipolemomentofthemolecule.ThestructureoftheSmoleculeisalsoshowntohighlightitspropellershapethatoptimizestheresultinglift(greyarrows) inherenttotherotationof the molecule in the fluid. Adapted from Ref. 27 with permission from the RoyalSocietyofChemistry.

The responding fraction of molecules F(E) can then beexpressed as the ratio ofmolecules following the rotation oftheelectricfieldoverthetotalnumberofmolecules(eq.16):

2 /2 2

( ) 1( 1)B

BB

E K TE E

F Ek Tk T e µ

µ µ»= -

-

(16)

whereµ andE are theabsolutevaluesof thedipolemomentandelectric fieldmagnitude.T is the temperature andkB theBoltzmannconstant.Assuminganelectricfieldmagnitudeof6× 105 V m–1 and a molecular electric dipole moment of 5.3Debye,theyestimatedthenumberofmoleculesrotatingwiththeelectricfieldtobeonly5moleculesoutof1000.Toverifytheirtheoreticalexpectations,thecapillarychamberwasfilledwith a solution of 1,1’-Bi-2-naphthol-bis(trifluoromethanesulfonate), which was submitted to arotating electric field (REF) for 83 hours and subsequentlyeluted out of the microfluidic channel. The eluted fractionswere analysed by circular dichroism (CD) and UV-visspectroscopy(Fig.11)showingenrichmentalongthecapillary,which isconsistentwiththepropeller theoryarguingthatthetwoenantiomersmoveinoppositedirections.





Furthermore,whentheREFisswitchedfromCWtoCCW,bothenantiomers reversed their migration direction. This studyconvincingly shows that chiral drift can compete withBrownianmotionatthemolecularlevel.

Fig. 11 Enantioenrichment of a racemic mixture of the 1,1’-Bi-2-naphtholbis(trifluoromethanesulfonate)molecules after 83 hours of REF. a) (Top) AbsorbanceandCircularDichroicchromatogramsofaracemicmixturewhileturningthefieldCCW(middle) CW and (bottom) CW for the pure S enantiomer. One can see on thechromatograms that the leading and the trailing fractions of the chamber displayoppositeCDsignalsthatisreversedwhentherotationdirectionofthefieldisreversed.(Bottom)WhenapureSenantiomer is loaded in thecellandsubmitted to thesameexperimentalcondition,bothfractionsdisplayaCDsignalofthesamesignwhich isacontrolexperimentforchiralmigrationobservedinthepreviousracemicexperiments.b) Absorption chromatogram of the chambers slug after being exposed a CW REFcollected from the in-linedetector. The slug canbedecomposed in twohalves (greyareas). The first left halve was enriched with the (S)-1,1’-Bi-2-naphthol bis(trifluoromethanesulfonate) whereas the second one was enriched with the R-analogueby61percent.AdaptedfromRef.27withpermissionfromtheRoyalSocietyofChemistry.

5ResolutioninmicrofluidicdevicesMicrofluidic devices are microchip-based integrated fluidiccircuits that can perform one or several (bio-)chemicallaboratory functions on a sub-millimetre scale. They offernumerous capabilities that are of great use in practice, likeeasy fabrication, high portability, the processing of tinyamountsofsamplesandreagents,highprocessingspeedsandshort reaction times, parallel operation, low costs etc.28Ultimately,manyadvances in the fieldofmicrofluidicdevicesare inspiredbythevisionof integratinganentirebiochemicallaboratoryontoasinglemicrochip(“lab-on-a-chip”).Forthesereasons, the miniaturization of classical, chemistry-basedenantioseparation techniques to the microchip format is arapidlyadvancingareaofresearch29,30sinceitsfirstrealizationaround 25 years ago29. Concerning chiral separation by

mechanical means, it turns out that microfluidic devicesprovide an ideal environment for theoretical developmentsandexperimentalproof-of-conceptstudies.

Fig. 12 a) Illustration of the two-dimensionalmodel for chiral particles proposed byKosturetal.31Sphericalcomponent“atoms”orchemical“groups”(symbolizedbythecoloured circles) of different size are rigidly connected (solid lines) to form a rigidparticleor “molecule”.Due thedifferent sizesof the individual “atoms”, sucha two-dimensional particle has a specific handedness. The two enantiomers of triangularshape particle are shown in blue and red; the dashed line indicates the mirrorsymmetryaxis.b)4-vortexflowpatterninthetwo-dimensionalx-yplane,employedbyKosturetal.31(andlaterbyBurgerandBeleke32,33)tostudychiralseparationinmicro-flows.Streamlinesofthefluidflow ( , ) ( / , / )x yv v y xy y= ¶ ¶ ¶ ¶ obtainedfromthestreamfunction( , ) sin( / ) sin( / ) / [2 cos( / ))(3 2 cos( / )]x y x yx y x L y L x L y Ly p p p p= - -

with 1x yL L= = are shown. Adapted from Ref. 31 with permission from theAmericanPhysicalSociety

5.1Vortexflows

The first proposal to use micro-flows for mechanical chiralseparationappearedin2006withatheoreticalstudybyKosturetal.31Theyconsideredanopenmicrofluidicsetuponaplanarpiezoelectricsubstrate, inwhichfluidflowisagitatedwithinathin liquid layer on the substrate by surface acoustic waves.This setup had been motivated by an earlier experimentaldemonstration34 that surface acoustic waves, whichcorrespond to a quadrupolar force density, create a flowpatternoffour,pairwisecounter-rotatingvorticesarrangedona rectangular lattice, see Fig. 12b. Kostur et al. used asimplified 2-dimensional model of this 4-vortex streamingpattern to investigate its effect on 2-dimensional chiralparticles. The chiral particles were constructed from threerigidly connected “atoms” with different friction coefficientsarrangedinatriangularpattern;seeFig.12a.Twoenantiomersofoppositehandednessarerelatedtoeachotherbyswitchingtwooutof thethree“atoms'' in the triangularsequence (it iseasy to see that this operation is equivalent tomirroring theparticle inthe2-dimensionalplane).Each“atom”experiencesa hydrodynamic friction force, which is proportional to itsvelocity relative to the fluid with a proportionality constantequal to the “atom's” friction coefficient. These particles

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therefore feel a mechanical force in a fluid flow, which issensitive to their chirality. Moreover, it is straightforward totake into account diffusive motion by adding thermalfluctuatingforcesintheequationofmotion.By extensive numerical studies, Kostur et al.31 demonstratedthat the purely advective motion (without thermalfluctuations) within each one of the vortices from Fig.12b isgoverned by a set of deterministic attractors, which ischaracteristic for the particle's chirality and the floworientationinthevortex.Undertheinfluenceofweakthermalnoise, almost all enantiomers of a specific chiralityconsequently settle in that attractor that has the higheststability for the given chirality. Hence, after reaching thestationarystate,thetwodifferenttypesoffluidvortices(withleft-handed or right-handed flow orientation, see Fig. 12b)contain different amounts of a specific enantiomer. In otherwords, Kostur et al.31 predicted from their model that anenantiomericmixturecanbeseparatedmechanicallybyusingfluidvorticesinamicrofluidicdeviceas“chiralselectors''.The model of Kostur et al.31 is based on a few quite drasticsimplifications. Most importantly, it is restricted to twodimensions, the chiral particles do not act back on thehydrodynamic flow field such that hydrodynamic interactionsbetween the component “atoms” are neglected, and thebonds between these “atoms'” are rigid, prohibiting particledeformations. Despite these simplifying assumptions, Kosturet al.31 argue that their model does capture the relevantaspectsofsymmetryandthermalnoiseinthemotionofchiralobjects. As for the first simplification, one could think aboutconfining the chiral objects to a surface or a liquid-liquidinterface17 in a way that also suppresses out-of-planerotations, such thateffectively the system is two-dimensional(seeFig.4above).Concerning hydrodynamic interactions, two follow-upstudies32,33 analysed the same 4-vortex flow pattern and itspotential for chiral separation employing two differentcomputational methods, which both allow for a fullhydrodynamictreatmentoftheproblem.Thefirststudy32usesa so-called fictitious domain Lagrange multiplier method,which has been developed specifically to capture thedeterministic motion of rigid particles in carrier fluids. Thesecond study33 uses a finite element immersed boundarymethod. Both techniques numerically solve the two-dimensional incompressible Navier-Stokes equations for thefluid velocity field in the presence of the immersed particlestogetherwiththeirrigidbodyequationsofmotion.Thevalidityof thismodel and the numericalmethods have been verifiedby comparing experimental trajectories with trajectoriesobtained from the numerical simulations for both, non-chiraland chiral particles. Left-handed and right-handed L-shapedenantiomerswereusedaschiralparticles,withtheirlong“leg”being about 300 𝜇m and their short “leg” about 200 𝜇m inlength. For such relatively large particles, the effects ofthermal fluctuations are negligibly small. Both studies32,33consistentlypredictedthattheseenantiomersareseparatedinthe4-vortexflowaccordingtotheirhandednesswheninjectedinto the flow approximately in the middle between two

counter-rotating vortices, because left-handed (right-handed)enantiomers are attracted by clockwise rotating (counter-clockwiserotating)vorticestocirclearoundtheircentres.Whileinthefiniteelementimmersedboundarymethod33theparticles are perfectly rigid, the fictitious domain Lagrangemultiplier method32 allows to even consider deformableparticles. Indeed, the simulationswereperformed for slightlydeformable L-shaped enantiomers, but did not showqualitative differences to the rigid particle simulations inBeleke-Maxwelletal.’swork.33Two earlier studies35,36 had investigated flexible chiralparticles, which are built of “atoms” connected by elasticsprings, when moving in three-dimensional uniform shearflows. For helical particles, Talkner et al.36 showed fromnumerical simulations that hydrodynamic interactionsbetween the component “atoms” (via the Rotne-Pragermobility tensors) lead to mechanically induced drift in adirection lateral to the shear plane, which depends on thechirality of the object and the sign of the shear rate. It eventurned out that the shear flow can deform achiral flexibleparticles into chiral shapes, which then experience a similarlateral drift motion. Such shear-induced chirality and lateralmigration had been reported by Watari and Larson35 forsymmetrically shaped tetrahedral objects with anisotropicrigidity. In both studies35,36 thermal noise effects had beenneglected.From thepractical viewpointof integratingmicrofluidic chiralseparationconceptsinto“lab-on-a-chip'”systems,easyaccessto the purified enantiomers for further processing orextractionisessential.IntheKosturetal.31proposal,particleswithspecificchiralityarespatiallyaccumulatedwithindefinedregionsofthe4-vortexflow.Aconvenientalternativemaybetomake the two chiral partnersmovewith different averagevelocities along a linear channel by imposing suitable flowpatterns.Inthisway,afirst“plug”enrichedinoneenantiomerarrives at the end of the channel before a second “plug”containing the other enantiomer,much like in themicrochipelectrophoretic versions of conventional separationtechniques29,30.Mechanicalchiralresolutionconceptsinlinearmicrofluidicchannelshavebeenstudieduptodate inseveralcontributions,boththeoretically37–39andexperimentally40,41.

5.2Microfluidicchannels

In a straight channel with a regular cross-sectional shape,pressure-driven flows develop a typical parabolic velocityprofile42. In particular, in a channel with a rectangular cross-sectionandahighaspect-ratio(width/height),theflowprofileisparabolicintheheightdirectionanduniformoverthewholechannel width. The corresponding shear rate varies linearlyover the channel height with negative (positive) sign in theupper (lower)halfof thechannel.Marcosetal.40 studied themotionofhelicallyshapedbacteriaofabout16𝜇minlengthinsuch a parabolic Poiseuille flow. They observed a lateral driftmotion perpendicular to the flow with a direction thatdependsonthechiralityofthehelixandthesignoftheshearrate, i.e. an effect similar to theoneobserved in rheometers





(see Sec. 3) and to theonepredicted for flexible particles bythe numerical studies ofWatari and Larson35, and Talkner etal.36 For symmetry reasons the lateral drift in the Poiseuilleprofile was oriented in opposite directions in the upper andlower half of the channel. The analysis of Marcos et al.40revealed that for the lateral drift to occur preferentialalignment of the helical bacteria with the streamlines of theflowisrequired.IfrotationalBrownianmotionduetothermalnoise dominates over the alignment effects due to the flow,the bacteria are randomly oriented and the latter driftdisappears. Marcos et al.40 estimated that with currentlyavailablemicrofluidics technology, shear ratesof theorderof107 s–1 can be generated, sufficient to induce lateral driftmotionofhelicalparticlesassmallas40nm.

Fig.13a) Illustrationoftheset-upsimulatedbyMeinhardt39usingdissipativeparticledynamics. The chiral helical particle (green object) is advected along the y-directionthroughamicrochannel(greyrectangularbox)withsquarecross-section(x-zplane)bya pressure-driven fluid flow, which is asymmetric due to different hydrodynamicboundaryconditionsonthefourchannelwalls.Therightpanelshowsprofilesofequalflow velocities across the channel. b) Separation of right-handed and left-handedhelicalparticles.Histogramsofdistancestravelledalongthechannelaxis (y-direction)inacertain time for left-handedhelices (darkshaded)andright-handedhelices (lightshaded)areshown.Resultsfromsimulations,whichtakefullaccountofhydrodynamiceffects (left part), are compared to a simple Brownian dynamics simulation withouthydrodynamics (right part). c) Distributions for left-handed (L) and right-handed (R)helices across the channel in the two cases with and without hydrodynamics. Thequantities𝜎and𝜏 representthebasic lengthandtime-units inthesimulationmodel.AdaptedfromRef.39withpermissionfromtheAmericanPhysicalSociety.

SincetheupperandlowerhalvesofthePoiseuilleprofileinthechannel inducedriftmotions inoppositedirections,aracemicmixture would separate into four quadrants in the planeperpendicular to the flow, with diametrically opposedquadrants containing the same enantiomer. However, bothenantiomers would still move with identical (average)velocities along the channel. The prospects of distinguishingthe two enantiomers by distinct migration velocities along amicrofluidic channel have further been investigated inReference37–39,41.The first two studies37,38 are based on the simple two-dimensional rigidparticlemodel introducedbyKostur et al.31Byanalyzingtheadvectiveanddiffusivemotionofsuchchiralparticles in two-dimensional linear channels in the low

Reynolds number regime, Eichhorn37,38 showed that particleswithoppositechiralitymovewithdifferentaveragevelocitiesifthe flow patterns break mirror symmetry. Specifically,Eichhorn37,38 considered two possibilities for nonlinear flowswithbroken symmetry: first, apressure-drivenparabolic flowprofile in a two dimensional straight channel superimposedwith a linear electroosmotic shear flow, which is created bydifferent slip velocities at the two channelwalls42 ; second, apressure-driven flowthrougha two-dimensionalchannelwithastraightandaperiodicallystructuredchannelwall,leadingtoa periodic nonlinear flow pattern with broken mirrorsymmetry. The theoretical analysis,37,38 showed that the twoenantiomers,whilebeingadvectedbythefluidflow,rotateinslightly different manners with different preferentialorientations.Moreover,themomentaryadvectionspeedofanenantiomerresultsfromthecombinedeffectofthelocalfluidflow acting on all the component “atoms'”, and thereforedependsontheparticle'sangularorientation.Theneteffectisan average transport velocity through the channel, which ischaracteristicforthespecificchiralityoftheparticle.The model used by Eichhorn37,38 is based on simplificationssimilar to Kostur et al.31: the hydrodynamic interactionbetween thecomponent “atoms”ofa rigid chiralparticleareneglected (see Fig. 12), and the analysis is restricted to twodimensions.Thatchiralseparationindeedcanbeobservedinafluid flowwithouthydrodynamic interactions implies that theseparation process is not a direct consequence of thehydrodynamic nature of the forces acting on the component“atoms”. Hence, it would occur under any suitable drivingforce, whose effect on these “atoms'” results in an overallchirality, such as an electric field applied on chiral particleswhich consist of “atoms”or “chemical groups”with differentcharges38. The two-dimensional setting, however, is quiteunrealistic as a description of real microfluidic channels.Nevertheless, the analysis of the 2-D model providesimportant insights37,38 in accordance with the conclusion ofKostur et al.31: as long as thermal noise does not dominateover deterministic forces,mechanical chiral resolution occursin force fields that break chiral symmetry (and thus havespecificparityor“chirality”themselves)byspatialvariationsofforce direction and magnitude on scales comparable to thesizeofthechiralobjects.Guidedbythese insights,Meinhardtet al.39 proposed a method for chiral sorting in a straightmicrofluidic channelwith quadratic cross-section. In order tocreate the required asymmetric flow profile from a pressuredifference applied to the channel ends, they suggested thatthe channel walls be endowed with different hydrodynamicboundary conditions (e.g. slip lengths) by chemicalfunctionalization43.Meinhardtetal.39studiedthemigrationofhelicalchiralparticlesinthechannel(seeFig.13)bydissipativeparticle dynamics simulations44, a mesoscale computationalmethodwhichtakesfullaccountofhydrodynamicinteractions.They demonstrated that helices of opposite chirality movewith different average speeds and thus can be separated. Bycontrasting simulations with and without hydrodynamicinteractions,Meinhardtetal.39couldshowthattheseparationeffect without hydrodynamic interactions, as predicted by

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Eichhorn37,38, is very weak, and that effective separation isactuallydrivenbyhydrodynamics(seeFig.13).An actual experimental separation of colloidal chiral particlesin a rectangularmicrofluidic channel has been performed byAristov et al41. Rather than breaking chiral symmetry of theflowprofilebyasymmetriccoatingsofthechannelwalls,theymade use of a topographical structure consisting ofperiodicallyarrangedridges(seeFig.14),whichwasknowntocreateahelicalflowfield45withacirculationdirectionfixedbytheorientationoftheridgesrelativetothepressuredropoverthechannel.

Fig. 14 a) Experimental set-up used by Aristov et al.41 The left panel shows theschematic illustration of particles of opposite chirality R (green) and S (blue). Theyconsist of three mutually perpendicular “legs” of equal length (about 6 𝜇m) assimplified representationsof realhelices. In the theoreticalmodel, thechiral colloidshavebeenrepresentedby7rigidlyconnectedbeads.Themiddlepanelshowsanfalse-coloured SEM micrograph. The right panel shows a schematic of the microfluidicchannel (sidewalls not shown)with periodically arranged oblique ridges (red)whichcreateaswirlingflowpattern(W=150𝜇m,H=115𝜇m,H-h=50𝜇m,b=40𝜇m,L=146𝜇m,and Lch= 1 m). (b) Arrival time distributions at various fixed positions x along thechannel axis from experiments (upper panel) and simulations (lower panel)demonstratingseparationofthechiralcolloidalparticlesinaright-handedhelicalflowfield.AdaptedfromRef.41withpermissionfromtheRoyalSocietyofChemistry.

Chiral colloidal “test” particles of several micrometer in sizewere fabricated from photoresist SU-8 (see Fig. 14). In theseparation experiment, a racemicmixture of left-handed andright-handedsuchparticleswas injected intothechannelandthendriventomovealongthechannelbyaswirlingflowfieldwith right-handed circulation direction. Aristov et al.41measured the concentration of each enantiomer at differentpositions along the channel and found that the right-handedparticle species with a chirality that matches the chirality oftheflowpatternwastransportedwithhighervelocitythantheleft-handed particles (see Fig. 14). Numerical simulations oftheexperimentalset-up,basedonaBrowniandynamicsmodelincluding hydrodynamic interactions via a grand mobilitytensor41, showed excellent agreement with the experimentalobservations. Detailed analysis of the experimental data andthe numerical data revealed that the right-handed particlewereaccumulatedinthechannelcentre,whereflowvelocityishigh,whiletheleft-handedparticlespreferablyvisitedregionswithlowflowvelocitiesclosertothechannelwalls.Aristovet

al.41 estimated that current microfluidics technology shouldallowtoseparateparticlesofabout120nminsizeinachannelwithabout3.2𝜇mcross-sectionaldimensionand3cmlength.

6Resolutioninnon-chiralmicrolatticesA quite different route towards chiral separation inmicrofluidic devices was explored by the theoretical study46and the experimental follow-up47. Rather than built-instructural chirality in the flowpatternor channel shape, theyexploited dynamical chiral symmetry breaking in non-chirallattice potentials46 or post arrays47. Conceptually, thisapproach builds on the observation of de Gennes18 (seeSection 2) that small chiral objects, when sliding down aninclined plane or floating over a liquid surface (see Fig. 2),shouldgenericallymove indirectionsslightlydifferent for thetwoenantiomers,providedthatthermalnoiseisnegligible.

Fig.15 Threeexamplesofchiralmoleculesconsistingofrigidlycoupled“atoms”(reddotsconnectedbysolid lines).Theshadedbackgroundrepresentstheperiodic latticepotential with identical periods along the basic directions 1e and 2e . (d-f) Typicalsingleparticle trajectories for themolecules shown in the correspondingpanels (a-c)and their chiral partners (red and blue lines). The trajectories were obtained fromnumericalsimulationsoftheequationsofmotion, includingthermalnoise.Thetiltingofthelatticepotentialalongthediagonaldirectionswitchesperiodicallybetweentwoamplitudes, indicated by the diagonal double arrows labelled ( )A t . Note that themagnitudeand the fractionof thedrivingperiod forwhich the two tilting forces areappliedneedstobeadaptedtoeachspecificchiralmoleculetoachieveseparation inoppositedirections46.Thebars indicate50 latticeperiods.Adapted fromRef.46withpermissionfromtheAmericanPhysicalSociety.

In the theoretical work46, Speer et al. studied the motion ofsmall chiral “molecules'” in a two-dimensional generic latticepotential,where the sizeof thechiralparticles is comparabletothelatticeconstantofthepotential.Asinthestudies31,37,38described above, the chiral objects were modelled by rigidlyconnectedcomponent“atoms'”withdifferentviscous frictioncoefficients. By numerical simulations, Speer et al.46demonstrated that under the influence of a static “tilting”force the two chiral partners generically move along distinctdirection through the periodic surface. They discovered thatthe separation effect is strongest if the tilting force A isorientedalongthediagonalsymmetrydirectionofthesquarelattice potential, because then the two enantiomers move





alongthetwo latticedirections, i.e. the 1e and 2e directions.Moreover, which one of the two enantiomers migrates inwhichdirectionwas foundtodependon theamplitudeof A ,suchthatbysuitableperiodicswitchingbetweendifferent A -magnitudes(atfixedorientation)thetwochiralpartnerscouldbemade tomove into opposite directions (see Fig. 15). Thischiral separationmechanismproved to be remarkably robustagainstthermalnoise46.In the experimental follow-up work47, the two-dimensionallatticepotentialwas realizedbya square latticeof cylindricalposts in a microfluidic channel (post radius around 3 𝜇m ,lattice constant 20 𝜇m, channel height 6 𝜇m). As chiral testparticles,Bogunovicetal.47usedleft-handedandright-handedL-shapedobjects48withlongandshort``legs''ofabout15µmand10µm(seeFig.16a).

Fig. 16 a) Schematic top-view (x-y plane) of the microfluidic channel containing aperiodicpostarray,inclinedbyanangle𝛼withrespecttothechannelaxis(x-direction).Fluidandparticlescanbeinjectedintothelinearchannelviatworeservoirsatitsends.Asteadyfluidflowthroughthechannel isgeneratedbyapressuredifferenceappliedto the two reservoirs.Themagnified inset showsanopticalmicrograph imageof thesquarelatticeofcylindricalposts,aswellastwoL-shapedparticlesofoppositechiralitycolouredinblue(Lparticle)andred(𝛤particle).b)Experimentaltrajectoriesobservedin the set-up from (a) with 𝛼=15° for several of the both chiral particles from (a),demonstrating chiral separation in the post array. Adapted from Ref. 47 withpermissionfromtheAmericanPhysicalSociety

The dimensions of themicrofluidic channel and the particleswere matched, such that the chiral particles could edgethrough the post array without being able to rotate aroundtheir long axis and switch chirality in that way. The chiralparticles were advected through the post array by a steadypressure-driven fluid flow. For this setup, Bogunovic et al.47performed extensive numerical simulations of a simpleBrowniandynamicsmodeltosystematicallyinvestigatethenetvelocity and direction of motion of the enantiomers as afunction of the inclination angle between the post array andtheflowdirection.Then,asuccessfulexperimentalseparationofleft-andright-handedL-shapedparticleswasconductedforaninclinationangle𝛼=15°andafluidflowvelocityoftheorderof100𝜇m/s,seeFig.16b.

7OpticalforcesIthasbeenrecognisedinrecentyearsthatlightcanbeusedtoexert discriminatory forces on chiral objects and that thismightformthebasisofnewschemesforchiralresolution.The earliest work on chiral optical forces seems to be atheoretical proposal put forward by Yong, Bruder and Sun in2007.49 These authors considered a three-level model of an‘oriented’ chiral molecule, illuminated by three overlappingbeams of light in a ∆-type configuration. The enantiomers ofthis molecule were shown to follow different trajectories inspace due to their interaction with the light. The chiraldiscriminationhereresultsfromthecyclicnatureofthe∆-typeconfiguration(whichisuniquelysupportedbychiralmolecules,owingtotheirindefiniteparity)togetherwiththeenantiomer-dependent signs of the Rabi frequencies that connect thethree levels.LiandShapiropresentedarefinedtheoryof thisphenomenon and offered their own schemes50,51. It wasarguedshortlythereafterbyJacobandHornberger52,however,that such approaches are not feasibly in reality: when therotational diffusion is accounted for, chiral discrimination iseliminated or else severely reduced. Also noteworthy is atheoretical proposal by Eilam and Shapiro53 to spatiallyseparate chiral homodimers from their achiral heterodimercounterparts,bypreparingthemindifferentinternalstatessothat they experience different optical forces in a given lightfield.Again, rotationaldegreesof freedomwereneglected. Itis interesting that theseearlyproposals49–51haveelements incommonwith a spectroscopic technique for chiralmoleculesdemonstrated recently by Patterson, Schnell and Doyle54, inwhichthreerotationallevelsareusedtodiscriminatebetweenenantiomers on the basis of the sign of the product of threetransitionelectric-dipolemomentcomponents.The first experimental demonstrationof a chiral optical forcewas reported by Cipparrone and co-workers in 201155,foreshadowedbythetheoreticalworkofRossandLakhtakia56as well as Guzatov and Klimov57. Cipparrone and co-workersshowed that left-handed cholesteric liquid crystal dropletswere either trapped in or expelled from optical tweezers,dependingonwhetherthetweezershad left-orright-handedcircular polarisation. This form of chiral discrimination occursbecause the droplet acts as a circular Bragg reflector: left-handed light is reflected whereas right-handed light istransmitted, resulting in different optical forces. Cipparroneand co-workers also reported on anisotropic chiralmicrospheres, which exhibited much more exotic dynamicsthantheirisotropiccounterparts.55ItisinterestingtonotethatacircularBraggreflectoractsasa‘chiralmirror’,reversingthespin of incident light whilst maintaining the helicity (incontrast,aconventionalmirrorreversesthehelicityofincidentlight while maintaining the spin). The associated torque wasobservedbyDonato58,Brzobohaty59andTkachenko60 inthreeindependentstudies.Thatfactthatthistorquecouldbeusedforchiralsortingwasfound independently by Tkachenko and Brasselet in 201361 .After which they went on to demonstrate the first opticalchiral sorter.62 Their device consisted of two counter-

SoftMatter REVIEW




propagating circularly polarised beams of light with oppositehelicities.Cholestericliquidcrystaldropletsofagivenchiralityflowing through the sorter were deflected depending on thehelicities of the beams whereas achiral droplets wereunaffected;seeFig.17.Itwasfound,moreover,thatthesorterwasabletodeflectleft-andright-handeddropletsinoppositedirectionsevenwhenthefrequencyofthe lightwasremovedfrom the Bragg window. The (smaller) effect here wasattributed to discriminatory refraction and scattering ratherthan Bragg reflection. Tkachenko and Brasselet concluded byproposing that the functionality of their sorter might beextended down to the nanometre (molecular) scale byexploiting functionalised chiral microparticles as ‘chiralconveyers’62.In recent years, theoretical work has focussed on nanoscalechiral objects, including chiral molecules. Canaguier-Durand,Genet and co-workers63 proposed a systematic descriptionofthe forces and torques that chiral light fields can exert onchiralobjects.Inthiswork,chiralobjectsweredescribedinthedipolar regime and time-averaged forces and torquesevaluated from the classical Lorentz law. This approachrevealedanew typeof force, stemming from themechanicalcoupling between the electric-magnetic dipole polarizabilitytensorand thequantities (densityand flow) thatcharacterizethe chiral content of a light field.64 Remarkably, the reactiveanddissipativecomponentsofthechiralforcearedetermined,respectively, from the real (rotatory power) and imaginary(circular dichroism) parts of the electric- magneticpolarizability tensor of the chiral dipole. This in turn impliesthat the direction of the chiral optical force directly dependson the enantiomeric form of the chiral object considered, aproperty that gives unexpected opportunities for proposingnewmechanisms for enantiomeric separation involving chirallight,bothinthefarandnearfields65–68.

Fig.17Achiralopticalsortercombininglaminarflowwithchiralopticalforces.Particlescanbedeflectedtotherightorleftdependingontheirchiralityandthepolarizationofthe light.𝝌 = 0 refers to non-chiralmedium and𝝌 = ± 1 to right/left-handed chiralmedia.AdaptedfromRef.60withpermissionfromSpringerNature.

Cameron and co-workers 69,70 explained, independently fromGenet et al. 63, that a gradient in the helicity (00-zilch or‘opticalchirality’)ofanoptical fieldcanbeusedtoacceleratethe enantiomers of a chiral molecule in opposite directionsandthatthis‘discriminatoryopticalforceforchiralmolecules’might be exploited to spatially separate the enantiomers ofchiral molecules, prepared as constituents of a dilutemolecular beam.A key featureof theCameronproposalwas

theuseofopticalbeamswithonlyasmallanglebetweentheirpropagationdirections,which formsahelicitygratingwithanadjustableperiod71. It is interestingtonotethat thepotentialenergy underpinning this optical force is the same as thatwhich governs optical refraction, with the chirality sensitivecontribution in particular giving rise to natural opticalrotation63,72.Ina seriesofworks73–79 ,Bradshawandco-workersproposedthat the same potential energy69,70,72 might be used tosignificantly modify the local enantiomeric excess of chiralmolecules in solution, by illuminating the solution with acircularlypolarisedbeamof light.Theviabilityof this schemeand the validity of the analysiswas debated in two differentworks.80,81Rukhlenko and co-workers82 considered two counter-propagatingplanewavesofoppositehelicity,allowingforthepossibilitythatthewavesmighthavedifferentfrequenciesand/ or intensities. Neglecting interference between the waves,these authors showed that the average optical force exertedon an isotropic chiral nanoparticle by the light field isabsolutely discriminatory to leading order, for appropriatechoices of the frequencies and intensities of the waves thatdepend in general upon the properties of the nanoparticle.The chiral discrimination here can be attributed to circulardichroism:thechiralnanoparticleabsorbsphotonsatdifferentratesfromthetwowavesandsoexperiencesanetforce.Forequal frequencies and intensities, this optical field isessentially that proposed by Canaguier-Durand and co-workers63andusedbyTkachenkoandBrasselet intheirchiralopticalsorter.62Rukhlenkoandco-workerswentontoanalysethe optical force-induced diffusion in their setup andconcluded that “although the proposedmethod is unsuitableforthedirectseparationofchiralmolecules,itcanstillbeusedto separate them if they are attached to chiral particleswithlarge(chiralresponse)”.Again,thisisreminiscentofthe‘chiralconveyer’ idea put forward by by Tkachenko and Brasselet62.Complementingsuchexperiments,Brasseletetal.83havebeenable to selectively displace chiral liquid crystal microspheresusing forces arising from optical helicity gradients, therebyimplementing the classical analogue of the molecular Stern-Gerlach chiral sorter proposed by Cameron and co-workers69,70. Recently,Kravets84etal.extendedexperimentalattempts to observe discriminatory optical forces exerted onindividualchiralobjectsbychiral lighttomany-bodysystems, demonstrating chirality-selective mechanical separation ofrandomly distributed assemblies of right-handed and left-handed chiral microparticles by optical means. This workopens the way for further experimental implementation byconsideringparticleswithsmallersizes.Progress on the first experiment aimed at measuring thediscriminatoryoptical force forchiralmoleculeswas reportedby Jones and co-workers85. These authors used a Poincarébeam of light, which has a spatially varying polarisationpattern, to try to induce a measurable local enantiomericexcess in a racemic solution. They obtained a null result,attributedtotheoverpoweringeffectsofdiffusionanddipoleopticalforces.





To move toward manipulating chiral matter at nanometerscales, it is important to be capable of spatially controllingsingle chiral nano-object in freestanding conditions and,simultaneously, recognizing their enantiomeric form. Crucialsteps in these directions have recently been taken,with newschemes, such as those proposed by Cheshnovsky and co-workers86 for sensitive CD extinction measurements onindividual chiral artificial nano-objects and nanocrystals.Schnoering et al. have optically trapped singlemetallic chiralnanopyramidsthatdisplayremarkablylargechiralresponses87.Exploitingafundamentalconsequenceoftheconservationlawof optical chirality proposed by Poulikakos, Norris and co-workers88,theydevelopedapolarimetricsetupthatperforms,inside theoptical trap, chiral recognitionof such singlenano-enantiomers87.Inthiscontext,itisimportanttomentionthat,as demonstrated by Dionne et al.89, chiral atomic forcemicroscopyprobescurrentlyopennewexperimentalpathsforrevealingandquantifyingchiralopticalforcesatthenanoscale.An aspect of chiral optical forces that requires furtherattention in theory is the rotational degrees of freedom. It isimportant to distinguish between an isotropic chiral object(considered in many theoretical studies so far) and anisotropically averaged object, resulting from anisotropicindividual components, such as a chiralmolecule. Theopticalresponse of a chiral molecule is strongly dependent on themolecule’s orientation with respect to the light. The chiraloptical force experienced by a chiral molecule will thereforediffer in magnitude and sign from the isotropically averagedprediction. This subtlety should be taken into account at thesingle-molecule levelandmightevenbeexploitedto increasethe efficiency of separation. This argumentation is againsimilartothatmadeinFig.4:byrestrictingrotationaldiffusionchiralmechanicalforcescanbecomemoredirectional,leadingtobetterresolution.Anotherinterestingaspectisthepossibilitygivenbythechiralelectric-magnetic polarizability tensor to induce linear-to-angularcrossedmomentumtransfers.Suchtransferscantakevariousdynamicalforms,fromlateraland“pulling”forces90–94to“left-handed”optical torques95,96.As theyaremediatedbythe chiralityof thedipole, these forces and torqueshave thepotentialtoyieldnewall-opticalschemestoachievemolecularresolution. For this reason, these perspectives certainlydeservefurtherstudy.

ConclusionsChiralresolutionofenantiomersisandwillremainofeminentimportance for our society. Despite established separationtechniquesbasedon, e.g., chiral column chromatography thesearch for newmethods to achieve faster andmore efficientresolution is still ongoing. Mechanical chiral resolution is apromising approach, and can be achieved using mechanicaland/or optical forces as shown in this review. However, inmostcases(rotational)diffusiondecreasestheefficiency,sincethechiralforcesarecoupleddirectlytotheorientationofthemoleculeorparticle.Futurestrategies to lower therotationaldiffusionusingasecondary‘aligningfield’(e.g.,byapplyingan

electric and/ormagnetic field) are therefore very interesting.The theoretical understanding of mechanical resolution issteadily improving and increasingly taking into accountdiffusionandhydrodynamicinteractions.Forfuturetheoreticalstudies,thereisstillroomforimprovementbyincludingchiralparticle–particle interactions for example, whichwill becomeimportant for industrial systems working at highconcentrations.Overall, thescepticismofachievingmolecularscale mechanical resolution is slowly withering, opening upavenues towards a new generation of chiral separationdevices.

ConflictsofinterestTherearenoconflictstodeclare.

AcknowledgementsAT acknowledges generous funding from Solvay, VM waspartiallyfundedthroughSATTConectusAlsace.RPCgratefullyacknowledgessupportfromTheLeverhulmeTrust(RPG-2017-048). SMB thanks the Royal Society for the award of aResearchProfessorship(RP150122).REacknowledgesfinancialsupportfromtheSwedishResearchCouncil(Vetenskapsrådet)under the grant No. 2016-05412. CG acknowledges AgenceNationalede laRecherche (ANR), France,ANREquipexUnion(GrantNo.ANR-10-EQPX-52-01),theLabexNIEprojects(GrantNo. ANR-11-LABX-0058-NIE), and USIAS within theInvestissements d’Avenir program (Grant No. ANR-10-IDEX-0002-02).

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Mechanical chiral resolution FINAL(with modifications of