Chiral Analysis and Separation Using Molecular Rotation

CHAPTER 19

Chiral Analysis and Separation Using Molecular Rotation Mirianas Chachisvilis Solvexa LLC, Keswick, CT, San Diego, CA, United States

19.1  INTRODUCTION Separation and analysis of chiral molecules also plays an important role in the pharmaceutical industry [1]. Most of the new small-molecule drugs reaching the market today are single enantiomers, rather than the racemic mixtures. Therefore, it is expected that separation and analysis of chiral molecules will continue to play an increasingly important role in the pharmaceutical industry [1,2]. More generally, the relevance of chirality in nature is well established [3]. A variety of mechanisms have been proposed to explain symmetry-breaking interactions and the origins of enantiomeric homogeneity in biological systems, such as circularly polarized light [4], gravitational fields and vortex motion, parity violation, time-dependent optical and magnetic fields, or photochemistry [5]. These mechanisms mostly lead to very small enantiomeric excess and thus require additional amplification to reach enantiopure state. Current separation methods typically rely on interactions with various chiral selectors, for example, chiral chromatography or recrystallization and related Viedma ripening [6,7], whereas determination of absolute configuration (AbCon) relies on X-ray crystallography and chiroptical spectroscopy [8], which encompasses a range of spectroscopic techniques, including vibrational circular dichroism [9], and typically requires ab initio simulations and/or large amount of sample and suffers from low fidelity. All these methods are time consuming and they do not lend themselves to a priori predictions of performance for newly synthesized molecules (e.g. chiral high-performance liquid chromatography requires a method development step such as selection of appropriate chiral column, solvent, and separation conditions). Other recently proposed chiral separation/analysis methods include chiral gratings [10] and nuclear magnetic resonance in the presence of static electric field [11,12]. Very recently, AbCon has been determined using microwaves [13] and Coulomb explosion imaging [14]; however, Chiral Analysis. http://dx.doi.org/10.1016/B978-0-444-64027-7.00019-5 Copyright © 2018 Elsevier B.V. All rights reserved.

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these new methods are applicable to molecules that can be sampled in the gas phase and have not been demonstrated for larger molecules with high degree of conformational freedom. In this chapter, we will describe and review two different chiral separation and analysis approaches that rely on molecular propeller effect (MPE). The macroscopic propeller effect is a well-known hydrodynamic phenomenon which manifests itself through rotational–translational coupling in left–right dissymmetrical bodies such as helical filaments [15–17]. Already Pasteur [18] claimed that dissymmetry generated by a rotation coupled with linear motion is similar to spatial dissymmetry (chirality) as encountered in chemical structures [3].Thus, chiral molecules can be envisaged as tiny propellers with their “handedness” and propulsion direction being determined by the AbCon of the molecule. It has long been hypothesized that the MPE may lead to separation of enantiomers exposed to radio-frequency electric fields of rotating polarization, however due to deficiencies in the theoretical derivations, the MPE was predicted to be small, which may have discouraged experimental verification of such an effect [19]. Recently, it has been experimentally demonstrated that the MPE is real and can be used to separate enantiomers of small molecules in solution using rotating electric field (REF) [20]; moreover, it was shown that the MPE can be used for AbCon determination. Rotating the molecules with electric fields requires a high electric dipole moment of the molecule, strong electric field, and nonpolar solvents [20]; such requirements may pose challenges for achieving commercial acceptance in the near future. Alternative approach is to use other possible means of imposing rotation on the molecules such as shear flows (SFs). Both approaches have their advantages and disadvantages as will be described in this chapter.

19.2  CHIRAL SEPARATIONS IN SFs There have been multiple theoretical [21–28] but only a few experimental studies [29–32] on separation of macroscopic chiral objects (such as helical colloidal particles, model chiral helices, helical-shaped bacteria, >1  µm) in helical flows, vortices, microfluidic SFs, or rotating magnetic fields; one of the more recent studies [30] points out that despite many theoretical approaches proposed so far, there is still no agreement as on the magnitude and even the direction of motion of chiral particles in SFs. Most of the theoretical studies presented in the above papers are phenomenological and

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rely on numerical simulations arriving at sometimes conflicting conclusions, although all agree that separation of chiral objects in SFs should be possible, at least on the microscale.The most detailed theoretical study by Makino and Doi [22] suggested that chiral separation for small molecules in SFs is possible but only at high shear rates because the separation effect is a third-order effect in the shear rate; the authors postulate that the effect must be zero at low shear rates (i.e. in the linear regime) due to random orientations of small molecules in solution (i.e. isotropic system). The magnitude of the shear rate . g relative to the rotational diffusion rate can be represented by dimension. less Péclet number, Pe, defined as: Pe = γ /Dr , where Dr = kBT / πηd 3 is rotational diffusion constant (η is the viscosity and d the size of the molecule). For small molecules discussed in this proposal, Pe < 10−3 even at the very high shear rate of 106 s−1 and therefore the random rotational diffusion dominates the rotational motion of small molecules even at high share rates. However, it is completely incorrect to assume that random distribution of molecular orientations always implies that the ensemble of the molecules will respond as an isotropic system. As we describe below and in our earlier paper [20], even for completely random distribution of molecular axes, the net rotational–translational coupling does not cancel out along an externally imposed rotation axis, thereby enabling chiral resolution.The key idea is that although the random diffusive rotational motions (due to thermal torques) are much faster than the rotation imposed by the SF (or by external electric REF [20], see below), these motions are independent (i.e. not correlated).That is the trajectories of individual molecules may appear completely random but their collective response can be significant and directional as long as symmetry considerations do not preclude it; we clearly demonstrate this point in a recent paper [20]. Therefore, and in contrast to the study by Makino and Doi [22], it is shown below that the chiral separation effect is linear in the shear rate even for small molecules (and small Pe numbers). Some of the other studies referenced above predicted that chiral resolution in achiral medium at molecular scale (1 nm) would only be possible on the time scale approaching that of universe [28]; our recent theoretical and experimental results show how flawed these earlier arguments were [20].The discussion below builds on the results and insights of our previous study in which we have developed the quantitative theory and methodology for calculation of rotational–translation coupling and the MPE (suitable for spatial scale <1 nm) based on the molecular dynamics simulations and the solution of rotational diffusion equation and confirmed the accuracy of the theoretical approach experimentally [20].

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We will focus on an effect of fluid shear on rotational motion of the particle. It has been long shown both theoretically [33–35] and verified experimentally [36,37] that a spherical particle placed in a Newtonian fluid in SFs will rotate around the axis perpendicular to the plane of shear at the frequency given by . γ ω = (19.1) 2 . where g is a shear rate; note that the rotation frequency does not depend on the particle size or solvent viscosity. Macroscopic ellipsoidal particles placed in SFs may undergo a more complicated periodic rotational motion following the so-called Jefferey’s orbits [35] with frequency . γ (19.2) ω= (rε + 1/ rε ) where rε is ellipticity defined as the ratio between the major and minor axes of the ellipsoid; however, on average, for microscopic particles of low-tomedium ellipticity (which applies to most small molecules relevant to pharmaceutical industry), Eq. (19.1) provides a good estimate of the expected average rotational frequency in the SF. It should be noted that Eq. (19.1) was derived assuming nonslip boundary condition which is confirmed to hold at the molecular level for most molecules dissolved in polar solvents based on fluorescence anisotropy decay experiments [38]. For the following discussion, we have selected 1,1′-binaphthyl-2,2′diamine molecule (Fig. 19.1 shows (S)-enantiomer) due to its similarity to two-bladed propeller and possibility of a simple analytical solution (molecule I). Based on symmetry considerations, it is expected that there will be a significant rotational–translational coupling around and along the axes I1 and I2 but not I3 (Fig. 19.1). Happel and Brenner [15] have provided a

Figure 19.1  (S)-(−)-1,1′-Binaphthyl-2,2′-diamine.

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simple analytical solution for the translational, rotational, and coupling tensors of a simple two-bladed propeller consisting of two thin circular disks connected by a thin rod (see Chapter 5.4 in Ref. [15]). Assuming that w is fixed and given by the Eq. (19.1), the propulsion velocity of such model propeller along the selected rotational axes I1 and I2 can be calculated by taking the ratio of the appropriate elements of the coupling and the translational tensors [15], respectively: d sin(θ / 2)cos(θ / 2) (19.3) υ1 = ω 2 2 + cos 2 (θ / 2) and d sin(θ / 2)cos(θ / 2) (19.4) υ2 = − ω 2 2 + sin 2 (θ / 2) where θ is the dihedral angle between the planes of the blades and d is the distance between the centers of the blades (θ = 90° corresponds to perpendicular blades). 2 = 0 due to the absence of rotational coupling along the I3-axis [15]. Notably, according to Eqs. (19.3) and (19.4), the propulsion velocity does not depend on the size of the blade or the solvent viscosity. However, it is clear that propulsion directions for rotation along the I1 and I2 axes are opposite due to opposite translation–rotational coupling. Fortunately, for any dihedral angle not equal to 0 or 90°, the magnitudes of 1 and 2 are different. Due to rotational Brownian diffusion, the orientations of molecular axes I1 and I2 are expected to be completely random with respect to the expected rotation axis (which is perpendicular to the plane of shear as noted above). Since I1 or I2 are not always perfectly aligned along the rotation axis, the propulsion efficiency is reduced by a certain factor which we call Acor. Acor can be determined by calculating the normalized averaged projection value of the relevant propeller axis (I1 or I2) on the rotation axis. As we show in our recent paper [20] for completely random distribution of molecular orientations Acor = 0.5. Therefore, the expected overall propulsion velocity can be calculated by taking the sum of the average projection values of 1 and 2 (and 3 in the general case) onto the rotation axis: (19.5) 〈υ prop 〉 = Acor (υ1 + υ2 + υ3 ) Since both 1 and 2 change sign for the propeller with opposite handedness, the prop value is opposite for the left-handed and right-handed propellers, thus, in principle enabling chiral separation in SFs as illustrated in Fig. 19.2. The most convenient method to produce laminar SF is

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Figure 19.2  Illustration of rotational direction and expected propeller motion for (S)and (R)-enantiomers of binaphthyl molecule in shear field of a circular Couette flow. The expected rotational axis is perpendicular to the drawing and pointing away from the reader.

Taylor–Couette system of two concentric rotating cylinders. Adjusting the gap between the cylinders and the rotation speed enables flexible variation of the shear rate magnitude. Such system has nearly homogeneous, linear shear field, and can support virtually unlimited molecular residence time needed to achieve the desired degree of chiral separation. As an example, for the binaphthyl diamine shown in Fig. 19.1, θ = 80.5° and d = 4.5 Å based on the structure optimized at ab initio B3LYP/cc-pVTZ level [39]. Then one can calculate the expected propulsion efficiencies Lrev1 and Lrev2 defined as an amount of displacement along the selected molecular axis after one full revolution along the same axis: Lrev1 = 2π 1/w and Lrev2 = 2π 2/w for I1 or I2 axes, respectively (identical definition as described in more detail in our paper [20]). For binaphthyl diamine, Lrev1 = 2.7 Å and Lrev2 = −2.88 Å. The overall propulsion efficiency that takes into account random orientations of the molecules with respect to the rotation axis can be calculated using the overall propulsion velocity value from Eq. (19.5) as 2π 〈υ prop 〉 (19.6) 〈L rev 〉 = ω For binaphthyl diamine molecule, 〈Lrev〉 = −0.092 Å per one revolution of the molecule. To verify whether the above propulsion efficiencies calculated using continuum hydrodynamic model are reliable, we have performed molecular dynamics (MD) simulations following the exact procedure described below and in detail in Ref. [20]. The simulations were performed in the explicit solvent (2490 acetonitrile molecules in 60 Å × 60 Å × 60 Å box with periodic boundary conditions, trajectory length of 125 ns). Random

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Figure 19.3  Results of molecular dynamics simulation of diamine binaphthyl molecule. Solid lines are the least-squares fit to the data; slope values are: around I1: 3.25 ± 0.03, I2: −3.35 ± 0.03, I3: 0.004 ± 0.02.

rotational motions (rotational diffusion) due to thermal torques acting on the molecule around selected molecular axes (Fig. 19.3) were used to plot the amount of displacement along the selected molecular axis versus the amount of rotation of the molecule around the same axis (see Fig. 19.3). Fig. 19.3 indicates that there is a correlation between the rotation around axis I1 or I2 and the propulsion along the same axis. As expected, there is no propulsion when the molecule rotates around the I3 axis. A linear fit to the data like shown Fig. 19.3 yields the propeller efficiencies, that is, the value of displacement per one revolution of Lrev1 = 3.25 ± 0.03 Å and Lrev2 = −3.35 ± 0.03 Å. In contrast, Lrev3 = 0.004 ± 0.02 Å for rotation around the I3 axis. It is apparent that the propulsion efficiency values obtained from MD simulations are remarkably close to those calculated using the simple hydrodynamic model described above. To test the effect of the solvent, we had performed MD simulations in other solvents, including cyclohexane, CCl4, and supercritical CO2 which

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indicated that translational–rotational coupling is within a factor of 20%, that is, the solvent molecule size or type does not significantly affect the value of Lrev. Note also that the above calculation method was verified experimentally to be quantitatively accurate in a recent paper on MPE for similar binaphthyl molecules [20]. The next step is to estimate what molecular rotational frequency can be reached at practically achievable shear rates. Realistically shear rates up to 106 s−1 have been reported for Newtonian fluids in narrow gap Couette flows [40,41] (and references therein); using this shear value, Eq. (19.1) yields the rotational frequency of 79.577 kHz. Based on Eq. (19.5), the overall propulsion velocity is then 0.73 µm/s, that is, counterpart enantiomers will be moving into opposite directions along the rotation axis at this velocity. To estimate the timescale and the degree of chiral resolution, we follow the approach described in our recent paper [20]. To quantify chiral separation efficiency, we use enantiomeric excess, ee (ee = 0% for racemic sample and ee = 100% for pure enantiomer). Using a model of diffusive spreading of two-overlapping Gaussian concentration profiles with opposite linear drift terms due to propeller effect (corresponding to the (S)- and (R)enantiomers moving in the opposite directions as illustrated in Fig. 19.4), it is straightforward to show that the enantiomeric excess is given by [20]:  〈v 〉 t  t (19.7) ee = erf  prop  ≈ 〈v prop 〉 πD D  2 Eq. (19.7) is derived assuming the sample is split into two halves at the center of the absorption profile. Intuitively, a square root dependence is indeed expected for low enrichment levels since a linear drift term due to propeller effect (v ⋅ t ) is acting against the diffusive spreading of the molecules (∼ 2Dt ). We will assume that the translational diffusion coefficient, D, for binaphthyl diamine is ∼1.5  × 10−5 cm2 s−1 based on the values obtained experimentally for similar binaphthyl molecules [20]. Assuming separation time, t, of 8 h, Eq. (19.7) yields 0.97% and 98.6% for enantiomeric excess and enrichment level, respectively. To compare expected performance for a different molecule, we have performed identical MD simulations as described above on S-(–)-2,2′(1,4-butylenedioxy)-6,6′-dinitro-1,1′-binaphthalene (molecule II) which is a larger molecule (MW of 430 versus 284 for binaphthyl diamine) containing an additional butylene linker that constrains the inter binaphthyl angle at a lower value of 70° (see the structure in Fig. 19.4 and in more

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Figure 19.4  Simulation of separation of counterpart enantiomers of (S)-(–)-2,2′-(1,4butylenedioxy)-6,6′-dinitro-1,1′-binaphthalene in acetonitrile at shear rate of 106 s−1 after 30 min. Dashed line indicates the profile of initially injected racemic sample at time zero.

detail in Refs. [20,42]); the analysis f MD trajectory results in the following propulsion efficiencies: Lrev1 = 0.19 ± 0.03 Å, 2 = −2.19 ± 0.03 Å, and Lrev3 = 1.22 ± 0.03 Å. The effect of the butylene linker can be attributed both due to the increase in the difference of the propulsion efficiencies when rotating around the I1 and I2 axes (because of the smaller dihedral angle) and due to the appearance of rotational–translational coupling along the I3 axis when compared to binaphthyl diamine (the linker acts as another, single “propeller blade” for rotation around I3). The overall average propulsion efficiency is again given by 〈L rev 〉 = Acor (L rev1 + L rev 2 + L rev 3 ) yielding a value of −0.39 Å per revolution. Using the same rotational frequency of 79.577 kHz, we then obtain from Eq. (19.6) the overall propulsion velocity of 3.1 µm/s. Eq. (19.7) then yields values of 0.98% and 99.2% for enantiomer excess and enrichment level, respectively, after only 30 min exposure to SFs (using D = 1.4 × 10−5 ± 5 × 10−7 cm2 s−1 obtained for this molecule by us earlier [20]). The chiral resolution is much faster for butylene binaphthyl mostly due to higher propeller efficiency but also helped by lower diffusion coefficient which reduces diffusive spreading (see Fig. 19.4).

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To evaluate the magnitude of the MPE for molecules with a single chiral center, we have performed MD simulations on (R)-(+)-Warfarin and (R)-(−)-Warfarin (a typical small drug molecule) in acetonitrile; preliminary analysis indicated propulsion efficiency of 0.85 Å/revolution which is similar to that of binaphthyls. The above estimates indicate that chiral resolution using SFs is possible and that the expected performance is improved over the method using Ref. [20] described below.

19.3  CHIRAL SEPARATIONS IN REFs In this section, we will discuss the case where molecular rotation is induced using REFs. To evaluate rotational–translational coupling in this case we selected two different binaphthyl molecules due to their suitable orientation of their dipole moments with respect to “propeller” axis (molecules II and III: 1,1′-bi-2-naphthol bis(trifluoromethanesulfonate)). When such a chiral molecule in solution is exposed to the REF, its dipole moment will tend to reorient and align with the electric field direction, leading to rotation of the molecule. With the macroscopic propeller analogy in mind, opposite enantiomers will propel in opposite directions along the chamber, leading to separation. Such an effect, similarly as discussed above, also offers a means of determining absolute configuration, provided that the direction of propulsion can be predicted theoretically. We introduce hydrodynamic chirality, to characterize the direction and magnitude of propeller-like motion of a molecule. The dipole moments of the selected binaphthyl molecules are parallel to the C2 symmetry axes (which is aligned along the I2 axis in Fig. 19.5A). An applied external electric field will impose a torque around any axis in the coordinate plane containing the I1 and I3 axes (Fig. 19.5A) depending on the spatial orientation of the molecule. Based on symmetry considerations, it is expected that rotation around the I1 axis will propel the molecule due to the propeller effect, whereas rotation around the I3 axis (aligned along internaphthyl bond) should not lead to significant propulsion. Due to random orientations of the molecules in solution with respect to the plane of the REF, the molecule on average will rotate around all possible axes in the I1 and I3 planes. Based on a simple hydrodynamic picture, the propulsion direction for a rotating binaphthyl critically depends on the dihedral angle between naphthyl moieties—similar to propeller blades. Therefore, the propulsion direction is expected to be of the opposite sign for different absolute configurations (S) and (R).

Figure 19.5  Theoretical results. (A) Binaphthyl molecule III ((S)-enantiomer). Dipole moment, µ = 5.3 Debye. (B) Mean square angular displacement of molecule III around three different molecular rotational axes. (C–E) Displacement of the center of mass of the molecule along a specified axis versus the angle of rotation around the same axis (only part of the trajectory is shown to reduce the number of points on the graph). Each data point represents a time step of 250 fs. (C, E) (S)-enantiomer, (D) (R)- enantiomer. Red lines indicate linear regression fit to the data. Slope values: (C) 1.22 ± 0.03 Å, (D) −1.18 ± 0.03 Å, (E) −0.02 ± 0.03 Å per 360° revolution (mean ± s.e.m.). (F) Distribution of molecules as a function of the angle between the external electric field and the dipole moment of the molecule (α), plotted for four different values of dipole moment–electric field interaction energies. Molecules above the dashed line represent the “responding” fraction of the molecules which respond to changes in the electric field direction. (G) The dependence of the expected propeller velocity on relative electric field magnitude calculated according to Eq. (19.4) for the following parameters: molecule rotation frequency, νeff = 0.9 MHz, displacement per one revolution, Lrev= 1.2 Å.

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To characterize rotational–translational coupling and estimate the expected amount of propulsion per one revolution of the molecule, we have performed MD simulations with the explicit solvent (benzene). Briefly, the optimization of the initial molecular structures and the calculation of dipole moments were performed at the B3LYP/cc-pVTZ level [43], using NWChem software package [39]. The electrostatic potential fit of atomic partial charges for molecules II and III was performed using CHELPG algorithm as implemented in NWChem. MD simulations were performed at 293 K using NAMD program (version 2.9) [44] and CHARMM general force field (CGenFF) [45]. The initial structure for the solute and solvent system comprised one solute molecule and 1463 benzene molecules in a 60 Å × 60 Å × 60 Å periodic box. As described above, random rotational motions (rotational diffusion) due to thermal torques acting on the molecule around selected molecular axes (Fig. 19.5B) were used to correlate the amount of rotation of the molecule around the selected axis with the amount of displacement along the same axis (Fig. 19.5C–E). Fig. 19.5C and D indicates that there is a correlation between rotation around the axis I1 and the propulsion along this axis; moreover, the direction of propulsion is opposite for the (S)- and (R)-enantiomers. As expected, Fig. 19.5E shows that there is no propulsion when the molecule rotates around the I3 axis. A linear fit to the data like shown in Fig. 19.5C and D yields the propeller efficiency, that is, the value of displacement per one revolution (Lrev) of 1.22 ± 0.03 Å and −1.18 ± 0.03 Å per revolution for the (S)- and (R)-enantiomers of molecule III, respectively. In contrast, Lrev = −0.02 ± 0.03 Å for rotation around the I3 axis. Similar MD simulations performed for molecule II yield Lrev values of only 0.19 ± 0.03 Å and −0.17 ± 0.03 Å (for (S)- and (R)enantiomers, respectively, as described above). This could be rationalized by the presence of an additional linker chain between the naphthyls that present a deviation from an ideal propeller shape.

19.3.1  Molecular rotation due to REF Next, we derive the dependence of the MPE on electric field magnitude (E), rotation frequency (ν), and electrical dipole moment of the molecule (µ). We assume that electric field rotation is much slower than the rotational relaxation time of the molecules (in low-viscosity solvents, typically in the 100–500-ps time scale [46,47]). Under this assumption, inertial effects can be neglected and the angular distribution density function, ρ(,θ,t) is given by the solution of the Smoluchowski equation in spherical coordinates [48]:

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∂U (ϕ ,θ , t )  D  1 ∂  ∂ρ 1 ∂  ∂U (ϕ ,θ , t )   = Dr ∆ρ + r  sin θρ + 2 ρ     . kT  sin θ ∂θ  ∂θ ∂ϕ  sin θ ∂ϕ    ∂t

(19.8) where ∆ =

1 ∂  ∂ 1 ∂ is the angular Laplace operator,  sin θ  + 2 sin θ ∂θ  ∂θ  sin θ ∂2 ϕ 2

U (ϕ ,θ , t ) = −µ(ϕ ,θ , t ) ⋅ E(ϕ ,θ , t ) is the dipolar interaction energy (µ and E are dipole moment and electric field vectors), Dr is the rotational diffusion coefficient), k is the Boltzmann constant, T is the temperature, θ is the azimuthal angle, and  is the polar angle. When the electric field rotates slowly, the distribution of molecules in angular space can be assumed to be always at equilibrium with any electric field orientation. The equilibrium solution of Eq. (19.8) is in the form of a simple Boltzmann distribution: e µ E c os(α )/kT ρ(α , E ) = (19.9) kT µ E /kT (e − e −µ E /kT ) µE where α is the relative angle between the electric field and dipole moment vectors; µ and E are the absolute values of dipole moment and electric field magnitude (since distribution is axially symmetric around the direction of electric field, there is no dependence on the azimuthal angle ). This bellshaped function exhibits a peak at an angle corresponding to the electric field orientation; Fig. 19.5F shows a two-dimensional projection of this function (for a time instant when electric field is parallel to any axis in a plane perpendicular to the separation chamber axis [20]).When the electric field rotates, the peak of the distribution function follows the electric field vector; however, its shape does not change provided molecules have sufficient time to adapt to the new direction of the field. Then the “responding” fraction of molecules (F(E)) can be calculated as a ratio of molecules that follow the rotation of the electric field (i.e. all molecules above the dashed lines in Fig. 19.5F) to the total number of molecules. To calculate F(E), we note at equilibrium the angular probability function of molecular dipole orientations is: P(θ , ϕ ) = e( µ E /kT )cos θ sin θ dθ dϕ where we assumed that electric field is aligned along the z-axis. Electric field vector is shown at the time instant when it is aligned along the z-axis, that is, |E| = Ez. Axial symmetry leads to the following integrated normalized molecular concentration density profile: z

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∫ ∫ dθ ∫

2π

e( µ E

z /kT

) cos θ

dϕ

e( µ E

z /kT

) cos θ

, which kT ( µ E / kT ) − ( µ E / kT ) e sin θ dϕ −e (e ) 0 0 µ Ez is equivalent to Eq. (19.9) when the electric field vector is aligned along the z-axis. The minimum in the dipole density is at θ = π, that is, when the dipoles are antiparallel to the electric field: µ Ez ρmin ( π , E z ) = . The total number of nonresponding 2 µ E z /kT kT (e − 1)

ρ(θ , E z ) =

π

0

2π

( µ E z /kT ) cos θ

dipoles is then: C non =

∫

π 0

=

z

z

ρmin ( π , E z ) sin θ dθ = 2 ρmin ( π , E z ) (these are

the dipoles below the dashed line in Fig. 19.5F that on average do not rotate in the rotation direction of REF). The total number of dipoles, C tot =

∫

π 0

ρ(θ , E z )sin θ dθ = 1 due to normalization.Therefore, the respond-

ing fraction is C − C non 2µ E z 2µ E F ( E z ) tot = 1− ≈ (19.10) 2 µ E /kT C tot kT (e − 1) kT z

(the subscript z can be dropped since the formula is valid for any orientation of the electric field). The responding fraction is an average fraction of the molecules that rotate following the REF. Assuming an electric field magnitude of 6 × 105 V  m−1 and a molecular electric dipole moment of 5.3 Debye (molecule III), F(E) ∼ 5 × 10−3 at room temperature, that is, on average only 5 molecules out of 1000 are rotating with the REF at any given time. However, it should be noted that all molecules rotate, both due to random thermal torques and the effect of the REF. However, the molecules belonging to a fraction below the dashed line (Fig. 19.5F) rotate equally randomly in both directions (as if they do not “experience” the electric field torque) and therefore they do not contribute to propeller propulsion. Whereas the fraction of molecules above the dashed line (a “responding” fraction) will rotate following the electric field rotation (in other words, the peak of the angular distribution will follow the electric field orientation), resulting in directional propeller motion mediated by rotational–translational coupling for those molecules. Alternative interpretation of F(E) could be that all molecules rotate but at a reduced, average frequency; for example, if field rotation frequency is 0.9 MHz, then all molecules rotate at 4.5 kHz (i.e. 0.9 MHz * F(E)). Note, however, this is not the effective rotational frequency we introduce

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below as there is an additional dynamic effect of molecules lagging if electric field rotational frequency is higher than molecules can follow. Eq. (19.8) is easily solvable analytically at equilibrium but not in the general, time-dependent case. In order to determine the effective rotational frequency of the molecule at a given rotational frequency of the external REF, we have solved the overdamped rotational motion equation with one degree of freedom which describes deterministic motion of a macroscopic dipole: dψ(t ) TE µ E (19.11) sin(2π t − ψ(t )) = = dt ξr ξr where TE is a torque imposed on the electric dipole moment by the field, ξr is the rotational friction coefficient determined from the Einstein relation ξ r = kT / Dr , and ψ(t) is the polar angle describing orientation of the dipole moment vector in a plane perpendicular to the chamber axis (for more details, see Ref. [20]). The term 2πt–ψ(t) in Eq. (19.11) represents the phase lag between the REF and the dipole moment. Eq. (19.11) can be easily integrated analytically:   ω ξr ψ(t ) = ωt − 2 tan −1   (19.12)  µ E + S cot(tS / 2ξ r )  where S = ω 2ξ r2 − µ 2 E 2 and w = 2πv. Fig. 19.6A shows the time dependence of the total dipole rotation angle (i.e. ψ(t)) at three different electric field magnitudes. These data illustrate that above a certain electric field strength, the molecule is capable of rotating with the REF, while at lower field magnitudes it “slips,” leading to an effectively lower molecular rotation frequency. As illustrated in Fig. 19.6A, the effective molecular rotation frequency is calculated as: v eff ( E ) = ∆ψ / 2π∆t . Fig. 19.6B shows veff as a function of the REF frequency ν; these results indicate that below v esc = µ E / 2πξ r (see below for more details on vesc), veff = v (and it does not depend on E), whereas at higher field rotation frequencies, veff falls off rapidly due to “slippage” effect, that is, the torque due to dipole–electric field interaction is not strong enough to rotate the molecule at the REF frequency. For example, at the field rotation frequency used in our experimental setup (0.9 MHz), the effective rotational frequency for molecule III is only 156 kHz, whereas for molecule II, veff = v (i.e. 0.9 MHz), mainly because the dipole moment of molecule II (10.9 Debye) is significantly larger.

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Figure 19.6  Effective rotational frequency. (A) The time dependence of the total molecular dipole rotation angle at three different electric field magnitudes. Parameters used: the REF frequency, ν = 0.9 MHz, rotational diffusion coefficient, Dr = 4 × 1012 deg2 s−1, dipole moment, µ = 5.3 Debye. (B) The effective molecular rotation frequency (veff) as a function of the REF frequency ν. Parameters used: v esc = µ E / 2 π ξ r = 0.507 MHz.

By taking the time derivative of Eq. (19.12) and assuming that field rotation frequency is much larger than vesc, one can obtain the following asymptotic analytical expression: v eff ( E ) ≈

2 v esc µ 2E 2 = 8π 2ξ r2 v 2v

(19.13)

 which indicates that when v >>vesc, veff is inversely dependent on the field rotation frequency and proportional to the dipole–field interaction torque squared. The velocity of a molecule due to propeller motion can be expressed as 

v = L rev × v eff ( E ) × Acor ( E ) × F ( E )

(19.14)

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where veff is an effective molecular rotation frequency and Acor is the angular correction factor. Excluding the Acor(E) × F(E) factor, Eq. (19.14) simply means that linear propulsion velocity is due to rotational–translational coupling. As indicated above, F(E) accounts for the fact that only a small fraction of molecules follow the REF, whereas the factor Acor accounts for the random orientation of the propeller axis. For weak electric fields, the dipoles diffuse out of plane perpendicular to the axis of the separation chamber. The electric field torque still causes these molecules to rotate around x axis, however the molecular propeller axis with highest rotational-translational coupling is not always aligned along chamber axis, hence the propulsion efficiency is reduced by a certain factor which we call Acor; Acor ≈ 0.5 for parameters used in this analysis (see Ref. [20] for details on calculation of Acor). Fig. 19.5G shows that for weak electric fields, propulsion velocity is linearly proportional to the rotation frequency and field magnitude (only in the case if field rotates slower than vesc). Also note that the linear dependence of propeller velocity on REF frequency Eq. (19.14) suggests that increasing frequency can more than compensate for the low F(E), enabling achievement of high-propulsion velocities and separation and/or analysis in a short time period. However, in Eq. (19.14), veff is equal to the rotation frequency of the REF only in the case when the molecular population is able to respond to a change in the orientation of the external electric field sufficiently fast, that is, it is able to follow the REF. The responding fraction of the molecules will be able to rotate at the frequency of the external REF if the maximal torque imposed by the dipole–field interaction is larger than the rotational drag torque due to the rotational friction experienced by the molecule in solution at the REF frequency. Above a certain REF frequency, the molecules are not able to follow the REF (see Fig. 19.6); the value of this “escape” frequency νesc is determined by the rotational friction, the dipole moment, and electric field strength. More specifically, νesc is a frequency at which the torque due to rotational drag is equal to the maximal torque imposed by the electric field on the molecular dipole: v esc = µ E / 2πξ r where ξ r = kBT / Dr is the rotational friction coefficient. vesc is ∼0.51 and ∼2.1 MHz for molecules III and II, respectively. At REF frequencies above νesc, the molecules “slip”, rotating at greatly reduced effective frequency veff (Fig. 19.6B). For example, for molecule I exposed to the REF of 0.9 MHz, veff is only156 kHz; in this case, Eq. (19.13) yields a velocity value of 25 nm s−1 at which molecule III is propelled in solution due to the propeller effect. The propulsion velocity of molecule II is larger

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at ∼45 nm s−1, mostly due to its much higher rotational escape frequency since Lrev for molecule II is significantly lower. Eq. (19.14) is only strictly valid when vesc is significantly higher than the frequency of the REF (i.e. when the assumption of equilibrium distribution Eq. (19.9) applies); nonetheless, our approximation based on introduction of veff is supported by experiment (see below). Note that when the field rotation frequency is much higher than vesc, veff ∼ E2 (see Eq. (19.13)); it then follows from Eq. (19.14) that the overall propeller velocity is proportional to the cube of the electric field magnitude. For this reason, our setup was designed to operate in the regime when the dependence on the electric field is nearly linear, that is, at higher voltage rather than higher frequency.

19.3.2  Experimental demonstration of MPE To experimentally confirm the MPE, we have performed experiments on solutions of molecules III and II using the experimental apparatus depicted in Fig. 19.7A. The REF inside the microfluidic chamber is generated by applying π/2 phase-shifted voltages to the four pairs of electrodes surrounding the chamber (Fig. 19.7B and C). A small amount of a racemic solution of molecule III was injected into the center of the separation chamber and exposed to the REF. Data in Fig. 19.8A show that the material collected from the leading and trailing sides of the exposed sample (which was split into two halves at the center of the absorption chromatogram) have finite and opposite signs of circular dichroism (CD) signal. If the rotation direction of the REF is inverted, the CD signals from the leading and trailing fractions are inverted too. Furthermore, when the experiment is performed on a pure enantiomer sample, no inversion of the CD signal for the leading and trailing fractions occurs; these results unequivocally prove that exposure to the REF leads to enantiomeric separation of binaphthyl molecules. Importantly, the experimentally detected direction of propulsion of (S)- and (R)-enantiomers is the same as predicted by MD simulations (compare signs of Lrev in Fig. 19.5C and D and CD signals in Fig. 19.8A and Ref. [20]; e.g. the leading fraction is enriched with the (S)-enantiomer for the clockwise REF, see Fig. 19.8A, middle panel). This offers a new approach for determining the absolute configuration of a chiral molecule. To quantify chiral separation efficiency, we use enantiomeric excess, ee (ee = 0% for racemic sample and ee = 100% for pure enantiomer) defined above. Based on the experimentally determined translational diffusion

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Figure 19.7  Experimental setup. (A) A three-dimensional slice of the separation chamber showing the four electrodes, A–D, surrounding the microfluidic capillary. (B) Cross-sectional schematic showing how the electric field rotates within the separation chamber at four selected time points during a single cycle, t1–t4. At each time point, two electrodes are in the high-voltage state (+), and the opposite two electrodes are in a zero-voltage state (0). This results in a 90° rotation of the orientation of the electric field (E) within the separation chamber between each time point. (C) The voltage waveforms on each of the four electrode pairs during one full cycle of the electric field rotation. These square-like waveforms (1100 V, 900 kHz) show the π/2 phase shift between electrodes A–D. The four time points t1–t4 from Fig. 2B are also shown. (D) Expected directions of motion of the (S)- and (R)-enantiomers of molecule III for the indicated direction of rotation of the REF (clockwise, curved black arrow). α is the relative angle between the electric dipole moment and the electric field. The electric field rotates around the x-axis in the plane zy. The gray arrows show the (opposite) directions of motion for the (S)- and (R)-enantiomers of molecule III. The structure of molecule III ((S)-enantiomer) is also shown with the dipole moment direction indicated.

Figure 19.8  Experimental results for molecule III. (A) Absorbance and CD chromatograms (obtained simultaneously) for samples of molecule III after being exposed to the REF for 83 h and subsequently collected. All process conditions were identical, except where noted. The first (left) peak of each chromatogram represents the leading half of the slug and the second (right) peak represents the trailing half of the slug. Upper: racemic molecule III after exposure to counter clockwise (CCW) REF. Middle: racemic molecule III after clockwise (CW) REF. Lower: pure (S)-enantiomer of molecule III after CW REF. (B) Absorbance chromatogram from the in-line detector of a slug of racemic molecule III after exposure to CW REF for 45 h. The sample collected from the shaded left-hand side of the chromatogram had enantiomeric excess (ee) of 26% of the (S)-enantiomer of molecule III, whereas the right shaded section of the chromatogram had an ee of 61% of the (R)-enantiomer of molecule III. This is consistent with the edge of the sample slug being more enantiomerically enriched than in the center. The boundaries of the active area of the separation chamber are also shown after conversion to the elution time scale.

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coefficient (molecule III: D = 8.3 × 10−6 ± 3 × 10−7 cm2 s−1; molecule II: D = 1.4 × 10−5 ± 5 × 10−7 cm2 s−1) and a drift term of 25 nm s−1, Eq. (19.7) predicts the ee value of 19% which is reasonably close to the experimental value of ∼26% (Fig. 19.8B) considering uncertainties in many parameters used to calculate v. Moreover, Fig. 19.8B indicates that enantiomeric enrichment is higher if the sample is collected from an off-center location of the chromatographic absorption profile, as compared to the ee value from one-full half of the sample (ee = 61%, corresponding to 80.5% enrichment level); this is expected for a diffusive spreading process with the presence of a drift term. We further tested if higher separation of enantiomers could be achieved using molecule II (Fig. 19.9A) which has a higher dipole moment (10.9 Da). CD chromatograms in Fig. 19.9B clearly show that after exposure to the REF, the leading and trailing fractions of the initially racemic sample become enantiomerically enriched. In contrast, when the experiment is performed on a pure enantiomer sample, no inversion of the CD signal from the leading and trailing fractions occurs. Moreover, and similar to molecule III, the sign of the CD signature depends on the direction of rotation of the REF, and the direction of translational motion is correctly predicted by MD simulation for (S)- and (R)-enantiomers, for example, the leading fraction is enriched in the (S)-enantiomer after exposure to the clockwise REF (compare CD signs in Fig. 19.9B, upper panel, and Supplementary Fig. 19.3 in Ref. [20]). Fig. 19.9C shows the concentration profiles after various exposure times to the REF, whereas Fig. 19.9D summarizes enantiomeric excess values as a function of separation time. These data indicate that enantiomeric excess is already detectable after 1 h of exposure to the REF. The enrichment curve in Fig. 19.9D exhibits a square root time dependence, as predicted by Eq. (19.7). This confirms that the underlying separation mechanism is linear in time, as expected for the propeller motion. A fit of Eq. (19.7) to the ee data in Fig. 19.9D using the experimentally determined value of D results in a propulsion velocity of ∼50 ± 5 nm s−1, which is very close to the theoretically predicted value (45 nm s−1, see above). The theoretical enrichment estimate for molecule II is 27% after 46 h which is also very close to the experimentally observed value of ∼32%. At higher exposure times, separation efficiency can deviate from the theoretical value primarily due to the finite length of the active area of the separation chamber (10 cm). The material that diffuses outside the chamber boundaries is not covered by the electrodes and therefore is not exposed to the REF.

Figure 19.9  Experimental results for molecule II. (A) Binaphthyl molecule II ((S)-enantiomer). Dipole moment, µ = 10.9 Debye. (B) Absorption and CD chromatograms (obtained simultaneously) for racemic molecule II, after being exposed to the REF for 21 h and subsequently collected. All experimental conditions were identical, except where noted. The first (left) peak of each chromatogram represents the signal from the leading half of the slug and the second (right) peak represents the trailing half of the slug. Upper panel: racemic molecule II after exposure to CW REF. Middle panel: racemic molecule II after exposure to CCW REF. Lower panel: pure (S)-enantiomer of molecule II after exposure to CW REF. (C) Three overlaid and normalized absorbance chromatograms from the in-line detector of racemic molecule II after exposure to CW REF for 8 min (blue), 6 h (red), and 60 h (black). The separation chamber boundaries, converted into the elution time scale, are also shown. The black chromatogram shows shape distortions due to effects from the chamber boundaries. (D) Enantiomeric excess (ee) versus CW REF exposure time for racemic molecule II. Each sample of molecule II was collected by splitting the sample at the center of the in-line chromatographic absorption profile into the leading and trailing halves, with the leading half being enriched in the (S)-enantiomer and the trailing half being enriched in the (R)-enantiomer. Each data point is an average of at least six samples (both (S) and (R) ee values averaged together), with standard error bars shown. The blue curve is the least-squares fit of Eq. (19.7) to the data.

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Our theoretical findings on the MPE are fundamentally different from an earlier theoretical study by Baranova and Zeldovich [19] which predicted a quadratic dependence on the electric field magnitude based on a phenomenological assumption that the propeller velocity should be proportional to the radio-frequency field intensity (∼E2). Moreover, they have assumed that a rate at which the molecules settle in response to the field change is faster than 100 MHz which as we show above is not correct for typical small molecules (i.e. they did not account for the fact that molecules would not be able to rotate at radio frequencies). Using the field magnitude of 3 × 105 V  m−1 and radio field rotation frequency of 100 MHz proposed in their study [19], veff would be only 308 Hz for molecule III, making experimental observation of the effect not practically feasible. In contrast, our derivation of Eq. (19.13) is based on the solution of rotational diffusion equation with two degrees of freedom, is valid for any electric field magnitude, and predicts that the propeller effect at practical electric field magnitudes will be multiple orders of magnitude stronger than those predicted by Baranova and Zeldovich, making experimental verification and potential applications feasible. For example, if we use the theory proposed in the Baranova and Zeldovich’s paper to calculate the propulsion velocity for molecule III in our experimental setup, we obtain a value of only ∼0.3 nm s−1 which is about two orders of magnitude lower than that predicted by our approach. As illustration if a solution of enantiomers is placed into a closed container of length L and exposed to an REF, over time an exponential concentration distribution will develop: vL (19.15) e( v / D ) x C ( x ) = C Ave D(e vL /D − 1) where CAve is the average concentration of enantiomers and D is the translation diffusion coefficient. The distribution profile is inverted for opposite enantiomers. Eq. (19.15) follows directly from the lateral diffusion equation with a drift term due to the propeller motion. Quantity D/v has a dimension of length and characterizes the balance between the diffusion and drift motion due to the propeller effect. A smaller characteristic length indicates that the propeller effect is stronger than diffusion, allowing a more complete separation of enantiomers within a shorter time period. Since the values of D/v are 3.4 and 3.0 cm for molecules III and II, respectively, it is expected that separation efficiency will be higher for molecule II.

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19.4  SUMMARY Theoretical and experimental evidence discussed above show that the strength of the MPE is underappreciated and can indeed enable separation of chiral molecules into enantiomerically enriched states within hours when exposed to REFs or SFs. The fields of drug discovery, development, and manufacturing increasingly require molecules that are enantiopure. In this chapter, we have described a novel approach for analysis and separation of chiral small molecules that utilizes MPE to enable simultaneous separation and absolute configuration analysis in a single step, without any need for chiral selectors. The technology may enable significant cost and time savings for chiral chemistry research and industry: (1) by eliminating the need for expensive chiral stationary phases, (2) by significantly shortening separation method development, and (3) through the use of predictive software for performance prediction and stereochemistry determination.

ACKNOWLEDGMENTS The work presented in this chapter was supported in part by the NIH grant R43GM119431 to Solvexa, LLC.

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