Molecules that only differ by their chirality, so called enantiomers, often
possess different properties with respect to their biological function.
Therefore, the separation of enantiomers presents a prominent challenge in
molecular biology and belongs to the ``Holy Grail'' of organic chemistry. We
suggest a new separation technique for chiral molecules that is based on the
transport properties in a microfluidic flow with spatially variable vorticity.
Because of their size the thermal fluctuating motion of the molecules must be
taken into account. These fluctuations play a decisive role in the proposed
separation mechanism

A. J. Lichtenberg

D. Figeys

J. K. G. Dhont

Marcin Kostur

Michael Schindler

Peter Hänggi

Peter Talkner

R. D. Astumian

S. Ahuja

English

arXiv

Crossref

Chiral Separation in Microflows

Molecules that only differ by their chirality, so called enantiomers, often possess different properties with respect to their biological function. Therefore, the separation of enantiomers presents a prominent challenge in molecular biology and belongs to the ``Holy Grail'' of organic chemistry. We suggest a new separation technique for chiral molecules that is based on the transport properties in a microfluidic flow with spatially variable vorticity. Because of their size the thermal fluctuating motion of the molecules must be taken into account. These fluctuations play a decisive role in the proposed separation mechanism

Kostur, Marcin

Schindler, Michael

Talkner, Peter

Hänggi, Peter (Prof. Dr. Dr. h.c. mult.)

OPUS Augsburg

arXiv:cond-mat/0506084 v2   24 Nov 2005Chiral separation in microflowsMarcin Kostur, Michael Schindler, Peter Talkner, Peter Ha¨nggiUniversita¨t Augsburg, Institut fu¨r Physik, Universita¨tsstrasse 1, D-86135 Augsburg, Germany(Dated: August 31, 2006)Molecules that only differ by their chirality, so called enantiomers, often possess different prop-erties with respect to their biological function. Therefore, the separation of enantiomers presents aprominent challenge in molecular biology and belongs to the “Holy Grail” of organic chemistry. Wesuggest a new separation technique for chiral molecules that is based on the transport propertiesin a microfluidic flow with spatially variable vorticity. Because of their size the thermal fluctuatingmotion of the molecules must be taken into account. These fluctuations play a decisive role in theproposed separation mechanism.PACS numbers: 47.85.Np, 05.60.Cd, 05.10.Gg, 05.40.JcThe main established methods of enantiomer separa-tion are gas or liquid chromatography and capillary elec-trophoresis [1]. In both cases specific chiral filling ma-terials are used to obtain different elusion times of theenantiomers. Moreover, long columns are needed throughwhich the enantiomers are dragged by a large pressure ora high voltage difference. Chemically less specific meth-ods that work with less substance and that do not requirehigh voltages or pressure would be of great advantage.During the last ten years the development of microfab-ricated fluid devices has experienced enormous progress[2]. New ideas how to manipulate small amounts of flu-ids and substances therein have been suggested [3]. Re-cently, it has been demonstrated that the agitation of flu-ids by surface acoustic waves on piezoelectric substratespresents a versatile method of manipulating and control-ling the flow of small amounts of a fluid [4, 5].Various separation mechanisms have been proposed.The separation according to size was predicted for par-ticles in a fluid that is periodically pumped through anarray of parallel cylindrical pores with ratchet shapedcylinder radii [6, 7] and experimentally corroborated inRef. [8]. de Gennes showed that a chiral crystal floatingon a liquid surface or gliding on a solid surface underthe influence of an external force will generally move ina direction that is specific for the chirality of the crystal[9]. For molecules, de Gennes argued thermal fluctua-tions would destroy this effect [9]. In the present letterwe propose a different mechanism that uses the construc-tive influence of thermal noise in combination with thenonlinear dynamics of molecules in a flow field to attaina separation of opposite chiral partners.Inertial forces acting on a suspended point-like parti-cle are generally small [10] and can be neglected. Conse-quently, the particle is advectively transported with thefluid velocity at the particle’s actual position. For anincompressible fluid, the particle motion then is volumeconserving, and, as a consequence, attractors do not ex-ist [11]. For small particles, diffusion provides anothertransport mechanism, which tends to level out concen-tration differences of suspended particles. In the case ofa point particle, diffusion in any incompressible flow re-sults in a homogeneous distribution if external forces likegravity or electric fields are absent.A contrasting picture results for extended particles.The local velocity of a surface point of an extended par-ticle need not coincide with the fluid velocity that onewould observe at this point in the absence of the par-ticle. As a consequence, the volume of the state spacespanned by the particle’s degrees of freedom is no longerconserved by the dynamics, which, as a result may dis-play attractors for stationary flow fields, in spite of thefact that the dynamics is reversible [12]. In general, sev-eral attractors will coexist. Which of them is approachedafter sufficiently long time depends on the initial condi-tions. If the particle is still small, say with a diameter of1 µm or less, thermal noise at ambient temperature willdestroy very weak attractors and populate stronger ones,with weights depending on the stability of the attractors.In order to illustrate the working principle, we studya model describing the motion of an extended planar ob-ject in a two dimensional incompressible stationary flow.This object, or “molecule” as we will call it, differs fromits chiral partners only in the sequence of three spherical“atoms” with different friction coefficients. For a chiralmolecule the mirror image does not match with the orig-inal molecule upon any motion in the plane. It is thedependence of the transport properties on the chiralitywhich is in the focus of this paper.At short times, the differences between the dynamics ofchiral partners possibly are rather small but at long timesthey will lead to different attractors with different stabil-ity properties. With the omnipresence of thermal noisein the fluid these different stability patterns will causespatial distributions that are different for chiral partnersto an extent that they can be used for an effective chiralseparation. This is a robust mechanism that also worksin three dimensions.Motion of an extended molecule in a flowing fluid.We consider three atoms with positions xi, i = 1, 2, 3that are rigidly connected, see Fig. 1. The force on theith particle exerted by the fluid moving with velocityv(x) is assumed to be proportional to the relative ve-locity of the particle x˙i(t) with respect to the fluid, i.e.,Ffli = γi (v(xi)− x˙i) where γi denotes the friction coef-ficient of the ith atom, and v(x) is the velocity field of2ϕ~r1x1 γ1 γ1~r2γ2x2 γ2~r3x3 γ3 γ3XdFIG. 1: A chiral molecule consisting of three atoms with dif-ferent friction coefficients γi on an equilateral triangle at po-sitions xi = X + ri(ϕ). The mirror particle differs only inthe sequence of the friction coefficients. The rigid rods (graylines) of lengths d maintain the structure.−11−1 1yxFIG. 2: The streaming pattern given by eq. (7) is shown in oneunit cell. The streamlines circulate within adjacent quartersin opposite directions.the fluid in absence of the atoms. In a fluid at tem-perature T , the frictional forces are always accompa-nied by random forces, which, for the sake of simplic-ity, are assumed to be independent Gaussian thermalwhite noises, ξi(t), with cartesian components ξi,α(t),α = 1, 2, satisfying the fluctuation-dissipation relation〈ξi,α(t)ξj,β(s)〉 = 2γikBT δi,j δα,β δ(t−s). We also have totake into account forces Fi,j caused by the jth atom andacting on the ith one that maintain the rigid structureof the molecule. We neglect hydrodynamic interactionsbetween the individual atoms [13] as well as small andvery short-lived inertial terms [10]. The particle motionthen follows from the balance of forces on each particleγi [v(xi)− x˙i] +∑j 6=iFi,j + ξi(t) = 0 (1)The actual degrees of freedom are translations of themolecule as a whole, and rigid rotations. The momen-tary position of the molecule is conveniently specified bythe center of friction X =∑i γ˜ixi where γ˜i = γi/∑j γjare normalized friction coefficients, and by an angle ϕfixing the orientation, see Fig. 1. The positions of theatoms can then be recovered asxi = X+ ri(ϕ) = X+ R(ϕ)r(0)i (2)where the vector ri(ϕ) pointing from X to the ith atomresults from a rotation R(ϕ) of a reference configurationr(0)i . The center of friction and the orientation then obeycoupled Langevin equations readingX˙ =∑iγ˜i v(X+ ri(ϕ)) + ξX, (3)ϕ˙ =∑iγ˜ir−2γ r′i(ϕ) · v(X + ri(ϕ)) + ξϕ (4)The dot and the prime denote derivatives with respectto time and angle, respectively. The length rγ =(∑γ˜ir2i (ϕ))1/2is independent of ϕ and represents an in-variant property of the molecule. The fluctuating forcesξX(t) and ξϕ(t) are linear combinations of the originalones. They vanish on average, are independent of eachother, Gaussian distributed, and therefore characterizedby their correlation functions reading〈ξXα(t)ξXβ(s)〉 = 2D δα,β δ(t− s), (5)〈ξϕ(t)ξϕ(s)〉 = 2D r−2γ δ(t− s) (6)where the noise strength is given by D = kBT/∑i γi.In distinct contrast to the two-dimensional velocityfield v(x), the three-dimensional vector field that gov-erns the motion of an extended particle generally has anon-vanishing divergence. It is therefore possible that inthe absence of the random forces the long-time dynamicsis ruled by one or more attractors which are approachedfrom different initial conditions. The actual structure ofthe deterministic dynamics will depend on the propertiesof the molecule and on the considered velocity field v(x).The velocity field. The velocity field of an incompress-ible two-dimensional fluid can always be expressed interms of a scalar stream function Ψ(x, y). For the sakeof definiteness we here choose a periodic stream functionΨ(x, y) =V0L0 sin(πx/L0) sin(πy/L0)(2− cos(πx/L0))(3− 2 cos(πy/L0)). (7)The velocity field follows as vx(x) = ∂Ψ(x)/∂y, vy(x) =−∂Ψ(x)/∂x. It is the divergence free solution of theStokes equation [13] for a fluid that is driven by aquadrupolar force density which we do not specify here.Within a unit cell (x, y ∈ [−1, 1]) it reproduces theproperties of the experimental streaming pattern in Ref.[4]. We use dimensionless variables with L0 = 1 andV0 =√15. The advective transport follows the lines ofconstant Ψ. Therefore, a unit cell contains four invariantquarters, each containing an eddy. Because the streamfunction transforms oddly under reflections at the co-ordinate axes, eddies in two adjacent quarters have anopposite parity according to the sense of rotation.3FIG. 3: (color). The deterministic motion of chiral partner molecules (top vs. bottom row) is visualized by the respectivedomains of attraction (left panels) and the corresponding attractors. The molecule has the shape of an equilateral trianglewith side-length d=0.2 and friction coefficients (γ1, γ2, γ3) = (1/6, 1/3, 1/2). Its chiral partner has the sequence (γ1, γ2, γ3) =(1/3, 1/6, 1/2). Their motions are determined by the deterministic parts of the eqs. (3, 4) with the velocity field (7). The leftcolumn shows cuts through the domains of attraction at ϕ = 2 within the upper right quarter of the unit cell, see Fig. 2. Therespective attractors are depicted in the right panels with corresponding colors. Trajectories starting from white regions do notapproach the indicated quarter. Note that the domain of attraction of the additional red attractor in the bottom row extendsbeyond the upper right quarter and consequently attracts trajectories starting outside of this quarter.Deterministic transport of finite particles. As dis-cussed above, the motion of finite, but still small, parti-cles is qualitatively different from the advective motion ofpoint particles. The deterministic dynamics in the three-dimensional state space spanned by the two translationaland one orientational degrees of freedom is no longer con-servative. It can be partitioned into different domains ofattraction. The detailed structure of the attractors andof their corresponding domains of attraction depends onthe size of the molecule, on the magnitude of the frictioncoefficients and also on their sequence and therefore onthe chirality of the molecule. for a pair of enantiomersFig. 3 depicts cuts of the domains of attraction at a fixedorientation ϕ and the corresponding attractors which areconfined to one quarter. In the particular case shown inFig. 3, we found three such attractors for one enantiomer,two of which are period-one attractors, i.e. after one fullrotation of ϕ the molecule’s center of friction has returnedto its initial position. The third attractor is chaotic. Incontrast, the respective chiral partner possesses four at-tractors in this quarter. Three of them are similar tothe attractors of the first enantiomer: The chaotic at-tractor and one of the period-one attractors still exist;the other period-one attractor is replaced by one withwinding number 42/43 (42 full rotations of ϕ correspondto 43 revolutions of X). The fourth attractor is peri-odic: Here, the orientation ϕ performs a libration andthe translational degrees of freedom move in relativelyclose distance from the boundary of the considered quar-ter. This and the chaotic attractors “collect” also pointsfrom adjacent quarters, see the left panel of the bottomrow in Fig. 3. In contrast, for the chiral partner only avery small set of points from outside the considered quar-ter is attracted, see the left panel of the top row in Fig. 3.For smaller molecules also more complicated trajectoriesexist that stay close to the boundaries of several quar-ters. However, also these trajectories differ for oppositechiral partners and populate preferentially quarters witha parity that is specific for the chirality of the molecule.Influence of noise. According to their different posi-tions and geometric structures, the attractors of unlikeenantiomers have different strengths within a quarter ofdefinite parity, and consequently differ in stability withrespect to thermal noise. At sufficiently weak noise al-most all molecules settle into the attractor with higheststability, provided the system is given sufficient time torelax into its stationary state. Fig. 4 depicts the resultingunequal distributions of a specific chiral molecule in quar-ters of different parity. For the transport of a moleculefrom a quarter with the “wrong” to one with the “fitting”parity the attractors close to the quarters’ boundaries,like the red one in Fig. 3, may act as a turnstile.Because transitions from any less stable to the moststable attractor will become increasingly rare with de-creasing noise, the approach to stationarity may pro-ceed slowly. On the other hand, with higher noise lev-els “wrong” attractors will be populated with increasingprobability and therefore the separation quality deterio-rates. In Fig. 5 the time to reach the stationary stateand the separation performance are compared as theyvary with the noise strength, which is quantified by thedimensionless quantity D/L0V0 = kBT/∑i γiL0V0. It4 1 0-1x 1 0-1y 2 1 0ρ(x,y)FIG. 4: For a single chiral molecule with size d = 0.2the stationary probability density integrated over all orien-tations ϕ exhibits a pronounced asymmetry with respectto quarters of different parity. The friction coefficients are(γ1, γ2, γ3) = (1/6, 1/3, 1/2) and the dimensionless noisestrength is D/L0V0 = 10−3.τ ≃ 104L0/V0τ ≃ 103L0/V0τ ≃ 102L0/V011.522.5N+N−2 · 10−4 10−3 10−2D/L0V0FIG. 5: The selectivity as given by the fraction of numbers ofmolecules with different chirality N+/N− in one quarter de-creases with increasing noise strength D/L0V0. The time τ itneeds to reach the stationary value increases with decreasingnoise strength. For the parameters of the molecule see Fig. 3.The error bars indicate the standard deviation as it resultsfrom the numerical simulation.is therefore not only determined by the temperature butalso by the total friction, as well as by the characteristiceddy size and the velocity scale. For example, to achievea selectivity of N+/N− = 1.5 within 15 min in water atroom temperature (viscosity η = 1g/cms) for this modelflow an eddy size of L0 ≈ 2µm is required. The size of themolecules then is fixed to d ≈ 0.4µm. This eddy size issmaller than what can presently be attained experimen-tally but it should be realizable in the near future [14].For this rough estimate we have assumed the validity ofthe Stokes law. With larger velocity gradients we alsoexpect to improve the separability for smaller objects.Conclusions. We elucidated the transport properties ofa chiral extended object in a two-dimensional model flowin view of efficient enantiomer separation. As a result,we propose to start from a uniform distribution of theracemic mixture and wait until the combined action ofthe advective transport and of the thermal fluctuationshas led to a stationary distribution. In this stationarystate, quarters with different parity will contain differentamounts of a specific entantiomer.There are several aspects already in two dimensionswhich we have not addressed with this investigation. Itis nevertheless clear that this model does capture the rele-vant aspects of symmetry and of the stochastic dynamicsof small but extended chiral particles. We are sure thatthese very aspects do also lead to chiral separation inmore realistic models describing e.g. spatially extendedatoms, hydrodynamic interactions between them, as wellas flexible bonds. Moreover, we are convinced that thesame mechanisms also work in a three dimensional heli-cal flow again leading to enantiomer separation. Surfaceacoustic waves provide a promising tool to produce suchflow patterns.AcknowledgmentsFinancial support by the Deutsche Forschungsgemein-schaft via the grants 1517/13, 1517/25 and SFB 486 B13is acknowledged.[1] S. Ahuja, Chromatography and Separation Science, Aca-demic Press, New York, 2002; C. Fujimoto (2002) Anal.Sci. 18, 19(2002).[2] D. Figeys and D. Pinto, Anal. Chem. 72, 330A (2000).[3] G. J. M. Bruin Electrophoresis 21, 3931 (2000); J. O.Tegenfeldt, C. Prinz, H. Cao, R. L. Huang, R. H. Austin,S. Y. Chou, E. C. Cox and J. C. Sturm, Anal. Bioanal.Chem. 378, 1678 (2004); H. A. Stone,A. D. Stroock andA. Ajdari, Annu. Rev. Fluid. Mech. 36 381 (2004).[4] Z. Guttenberg, A. Rathgeber, S. Keller, J.O. Ra¨dler, A.Wixforth, M. Kostur, M. Schindler and P. Talkner, Phys.Rev. E 70, 056311 (2004).[5] Z. Guttenberg, H. Mu¨ller, H. Habermu¨ller, A. Geisbauer,J. Pipper, J. Felbel, M. Kielpinski, J. Scriba and A. Wix-forth, Lab on a Chip 5, 308 (2005).[6] C. Kettner, P. Reimann, P. Ha¨nggi and F. Mu¨ller, Phys.Rev. E 61, 312 (2000).[7] R. D. Astumian and P. Ha¨nggi, Physics Today 55 no.11,33 (2002); P. Ha¨nggi, F. Marchesoni and F. Nori, Ann.Phys. (Leipzig) 14, 51 (2005).[8] S. Matthias and F. Mu¨ller, Nature 424, 53 (2003).[9] P. G. de Gennes, Europhys. Lett. 46, 827 (1999).[10] E. M. Purcell, Am. J. Phys. 45, 3 (1977).[11] A. J. Lichtenberg, M. A. Liebermann, Regular andChaotic Dynamics, Springer Verlag, Berlin, 1992.[12] A. Politi, G. L. Oppo and R. Badii, Phys. Rev. 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Chiral separation in microflows

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Chiral separation in microflows

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