Paul+DeGregory+-+Final+Paper

=An Overview of the Chiral Selectivity Via Inclusion Complexation and Separation Optimization Offered by Cyclodextrins for the Separation of Enantiomers in Capillary Zone Electrophoresis=


 * __Abstract__**

Cyclodextrins are cyclic macrocycles composed of glucose units and are used for a variety of applications ranging from catalysis to separation science. Relatively recently, the use of cyclodextrins in separation science has significantly increased as various cyclodextrins have been found to separate a large variety of enantiomers. Furthermore, the high efficiencies of capillary zone electrophoresis (CZE) and the ease of application of cyclodextrins to the separation technique have popularized the use of cyclodextrins in enantioseparations. The aim of this paper is to discuss how cyclodextrins are able to separate enantiomers and the options available for performing and optimizing enantioseparations involving cyclodextrins in CZE. Based on performed research, it can be concluded that cyclodextrins are able to separate enantiomers by forming inclusion complexes with enantiomers in which there are different binding constants for the complexation of each enantiomer with the cyclodextrin and different solvation layers between each enantiomer-cyclodextrin complex. Additionally, the use of cyclodextrins in enantioseparations in CZE allows for a variety of means by which the separations can be optimized ranging from changes in the ionic strength of the background electrolyte, pH, temperature, and applied voltage and the use of different cyclodextrins, different concentrations of cyclodextrin, polar organic solvents, multiple cyclodextrins, and ligand exchange mechanisms.


 * __Introduction__**

Separation of enantiomers has become an important topic in pharmaceutical analysis and bioanalysis for the screening of drugs, drug impurities, synthetic precursors, side products, and metabolites [1]. The reason for this is that both enantiomers of a chiral product are often formed in laboratory syntheses and often display different biological effects from one another [8]. Additionally, specific methods for the separation of enantiomers are needed as enantiomers display the same physical and chemical properties and possess identical electrophoretic mobilities [19]. The separation of enantiomers via the use of cyclodextrins in CZE is the most effective means of separating enantiomers by capillary electrophoresis [7]. Cyclodextrins are preferred for their use in enantioseparations as they are transparent to UV light, enabling commercial UV detectors to be employed without the cyclodextrins interfering with the output electropherogram. Additionally, cyclodextrins have a hydrophobic cavity, which enables them to be used to separate many enantiomeric drugs, which are typically hydrophobic [13].

Capillary electrophoresis is a preferred technique for enantioseparations for several reasons. Generally, capillary electrophoresis has higher plate numbers and selectivity than does HPLC [18]. Capillary electrophoresis also does not require the use of complex stationary phases as does HPLC [4]. Also, as GC can only analyze substances that can be practically volatized, capillary electrophoresis permits for the analysis of a greater variety of analytes than does GC. Capillary electrophoresis is also advantageous in that it requires small amounts of materials, such as analytes and cyclodextrins, which could be expensive or unavailable in large amounts [18].


 * __Structure of Cyclodextrins__**

Cyclodextrins contain α(1→4) linked D-glucopyranose units and can be produced by CGTase in bacteria. Cyclodextrins have a structure similar to the shape of a torus, contain a hydrophilic outer surface, and a lipophilic cavity. The wider rim of the structure contains secondary 2- and 3-hydroxyl groups while the narrower rim of the structure contains primary 6-hydroxyl groups. The most abundant cyclodextrins, which are found in nature, include α-cyclodextrin, β-cyclodextrin, and γ-cyclodextrin, which contain 6, 7, and 8 glucose units, respectively [9].


 * __Inclusion Complexation – Steric and Thermodynamic Factors__**

The ability of a cyclodextrin to form an inclusion complex with a substrate or guest molecule in which the cyclodextrin encircles and binds to a part of the guest can be explained by steric and thermodynamic factors. The steric factors involved depend on the relative size of the cyclodextrin to the size of the guest molecule and certain key functional groups attached to the guest. If the guest's size is too large or too small, the guest molecule will not fit into the cyclodextrin. For an inclusion complex to be thermodynamically favorable, there must be a favorable net energetic driving force that pulls the guest into the cyclodextrin. A favorable net energetic change for the formation of an inclusion complex could involve several processes. The release of polar water molecules from the apolar cyclodextrin cavity, the formation of a larger amount of hydrogen bonds between water molecules that are released from the cavity, a decrease in the amount of repulsive interactions between hydrophobic guests and an aqueous environment, and an increase in hydrophobic interactions as the guest moves into the apolar cyclodextrin cavity can contribute to a favorable net energetic change for the formation of an inclusion complex. Various explanations for the favorable energy changes that occur in the formation of inclusion complexes ultimately show that the complexation mechanism is neither simple nor universal for all combinations of cyclodextrins, guests, and solvents. Complex formation is usually associated with a large, negative ∆H and a ∆S that can either be positive or negative. However, classical hydrophobic interactions, which are based on the hydrophobic effect, are associated with a slightly positive ∆H and a large, positive ∆S and have been shown to drive cyclodextrin complexation in certain cases [1].

The principal driving force for complexation in aqueous environments is thought to be the release of water from the cyclodextrin cavity. Water molecules inside a cyclodextrin cavity are unable to satisfy their hydrogen bonding potentials and have higher enthalpies than they would have if they were outside of the cavity. When a guest that is less polar than water replaces water in a cyclodextrin cavity, the replaced water releases enthalpy, which lowers the system’s energy [1].

Other factors involved in complex formation include the release of ring strain in the glucopyranose units composing the cyclodextrin, hydrophobic interactions, van der Waal’s forces, hydrogen bonding, and the flexibility, or degrees of freedom of a guest [1, 2]. The relationship between ring strain and complex formation has been observed in the case of α-cyclodextrin. When α-cyclodextrin is associated with water, hydrogen bonds between the cyclodextrin and water perturb the cyclodextrin's structure. Removing the included water molecules eliminates the hydrogen bonds between the cyclodextrin and the water molecules and thus releases ring strain in the glucopyranose units. The relation of binding constants that describe the dynamic equilibria between cyclodextrins, guests, and cyclodextrin-guest complexes to substrate polarizability suggests that van der Waal’s forces are important in complex formation. Additionally, the use of adamantanecarboxylates as guests and α-, β-, and γ-cyclodextrins as hosts demonstrates the roles of van der Waal’s forces (characterized by a negative ∆H and a negative ∆S) and classical hydrophobic effects in complexation. For the use of α-cyclodextrin, experimental data showed small changes in ∆H and ∆S, which is indicative of minimal interaction between the bulky guest and the small cavity of the cyclodextrin. For the use of β-cyclodextrin, a large, negative ∆H and a near-zero ∆S were observed. These results suggest that a snug-fitting complex was formed and that van der Waal's forces were present in the complexation. Complexation with γ-cyclodextrin resulted in near-zero ∆H values and large, positive ∆S values, which likely resulted from consequences of the hydrophobic effect and a lack of van der Waal’s interactions. The lack of van der Waal's interactions can be explained by the large size of the γ-cyclodextrin cavity [1]. Regarding the effect of the flexibility of a guest on complex formation, as the rigidity of a guest due to certain moieties such as double bonds increases, the amount of conformations the guest could assume that would allow it to fit into a cyclodextrin’s cavity decreases. This is due to that complexation of a more rigid guest would be less entropically favorable [2].


 * __Thermodynamics of Cyclodextrin Enantioselectivity__**

The complexation of enantiomeric pairs of guest molecules with cyclodextrins typically involves small differences between the thermodynamic quantities describing the complexation of each enantiomer. For example, differences in ∆G° that are typically less than 1 kJ/mol are reported for the complexations of mono(6-anilino-6-deoxy)-β-cyclodextrin with enantiomeric pairs of various amino acids and for the complexations of α- and β-cyclodextrins with enantiomeric ephedrines and pseudoephedrines. Often, enantioselectivity is attributed to a larger ∆S° value for the complexation reaction involving the preferred enantiomer compared to the complexation reaction involving the less preferred enantiomer. The larger ∆S° value is a consequence of greater conformational freedom of the favored enantiomer inside of the cyclodextrin cavity and the classical hydrophobic effect in which there is more extensive desolvation of the cyclodextrin cavity in the system with the preferred enantiomer [2].


 * __Basis for the Enantioselectivity of Cyclodextrins__**

The five chiral carbon atoms in each glucopyranose unit are the source of the enantioselectivity of cyclodextrins [4]. Additionally, as cyclodextrins are shaped like tori, the shapes of the glucose units do not repeat from unit to unit. As a result, different chiral recognition sites are present between glucopyranose units of the same cyclodextrin. Furthermore, cyclodextrins can change their shape upon interacting with analytes, in which they can experience included-fit interactions, which can broaden the chiral selectivity of cyclodextrins [8].

The “three-point interaction” model describes how the interactions between the cyclodextrin and enantiomers are different for each enantiomer in a pair of enantiomers. This model states that the binding site between the cyclodextrin and each enantiomer contains three mutually independent interactions between functional or structural groups of each enantiomer and the cyclodextrin. Essentially, the three points of interaction at the binding site favor one enantiomer more than the other, resulting in enantioselectivity. The interactions can be attractive or repulsive, although at least one interaction must be attractive for the enantiomers to bind to the cyclodextrin, and can involve electrostatic repulsion, electrostatic attraction, steric hindrance, hydrogen bonding, hydrophobic interactions, charge transfer, π-π electron interactions, and dipole-dipole interactions [6].

Overall, the higher affinity of one enantiomer for the cyclodextrin is due to a more favorable steric orientation of the favored enantiomer in the cyclodextrin cavity [13]. However, due to the flexible structures of cyclodextrins their selectivity values are rarely greater than two. Compensation for the low selectivity values is achieved in CZE by the high efficiencies of the capillaries used [8].


 * __Direct and Indirect Enantioseparations with Cyclodextrins__**

Enantioseparations involving cyclodextrins can be categorized into indirect and direct separations. The basis for indirect separations involves the formation of enantiomer-cyclodextrin complexes that are diastereomeric to one another and that do not dissociate over the course of the separation. Differences in the shapes and volumes of the complexes and shapes and volumes of the solvation layers of the complexes are the bases for the difference in the mobilities of the complexes and the resolution of the enantiomers. The basis for direct separations involves rapid, dynamic equilibria for the complexation reactions involving the enantiomers and the cyclodextrin over the course of the separation [6]. For direct separations, the difference in the binding constants of the enantiomers with the cyclodextrin almost completely determines the selectivity of the separation. This is because the difference in binding constants causes the enantiomers to spend different portions of time complexed and free and an enantiomer has a different mobility when complexed than when free [6, 19].

With certain combinations of enantiomers and cyclodextrins the separation mechanism is direct at lower concentrations of cyclodextrin and indirect at higher concentrations of cyclodextrin. In these cases, the enantiomer that has the higher binding constant with the cyclodextrin elutes second in the direct mechanism and first in the indirect mechanism. This enantiomer would elute first in the indirect mechanism as the complex with the enantiomer would have a higher charge density due to the enantiomer being positioned deeper in the cyclodextrin cavity [3].


 * __Predicting the Success of Enantioseparations with Cyclodextrins__**

In general, for cyclodextrins to be used as chiral selectors the target enantiomers should have hydrophobic moieties located close to a chiral center as the hydrophobic cavities of cyclodextrins interact with the chiral centers of enantiomers [13]. In CZE, separations are based on the difference in the electrophoretic mobilities of the analytes. Since the mobilities of enantiomers are identical, chiral additives, such as cyclodextrins, are needed in separations [7]. Several factors affect the mobilities of the enantiomers in systems with cyclodextrins. Firstly, increasing the difference in the binding constants for each enantiomer with the cyclodextrin increases the difference in the mobilities of the enantiomers. Secondly, cyclodextrin concentration affects the mobility difference between enantiomers such that there is an optimum concentration, [C], of a cyclodextrin for a pair of enantiomers. An inverse relationship exists between the affinity of the enantiomers for the cyclodextrin and the optimum concentration of cyclodextrin. Thus, increasing the affinities of the enantiomers for the cyclodextrin decreases the optimum concentration of cyclodextrin. Thirdly, the difference in mobilities of the enantiomers is maximized when the mobility of the enantiomer-cyclodextrin complex is in the opposite direction to the mobility of each of the enantiomers. Also, knowledge of the binding constants is of great use since in addition to [C] the effective mobility difference between the enantiomers as a result of complexation phenomena with a cyclodextrin at [C] can be calculated [5].

Furthermore, a model has been constructed to predict the effective mobilities of enantiomers in direct separations based on 1:1 enantiomer:cyclodextrin interactions in open tubular capillary electrophoresis, which includes CZE. It has been demonstrated that the mobility of the free enantiomers, the mobility of the enantiomer-cyclodextrin complex, which is assumed to be the same for both enantiomers, the binding constants for the interactions between the cyclodextrin and enantiomers, and a viscosity correction factor can be used to predict the effective mobilities of enantiomers. Overall, experimental values have been shown to correspond very closely with predicted values from the model [17].


 * __Optimization of Enantioseparations in CZE with Cyclodextrins__**

For an enantiomer than can be separated by cyclodextrins, optimization of the selectivity and speed are the major goals, in which selectivity is prioritized over speed. However, in chiral separations, ways to improve selectivity and speed often contradict one another. Additionally, the most effective way to achieve higher selectivity in chiral separations is to increase the difference between the effective mobilities of the enantiomers, which is strongly affected by the concentration of the cyclodextrin [6]. Overall, there are several parameters, which are discussed below, that affect the ability to separate enantiomers and the optimization of such separations using cyclodextrins in CZE.

//Choice of Cyclodextrin//

Overall, there are no rules that lead to the selection of the most appropriate cyclodextrin for the separation of enantiomers as theories that satisfactorily describe the interaction mechanisms of cyclodextrins with various chiral compounds do not exist. Estimation of the capability of a cyclodextrin to discriminate between chiral compounds based on separations of closely related compounds is often employed but is of limited reliability [6]. However, as the use of different cyclodextrins affects the binding constants of the enantiomer-cyclodextrin complexes, the affinities of the enantiomers for the cyclodextrin, the difference between the mobilities of the enantiomer-cyclodextrin complexes, and the difference between mobilities of the enantiomers, the potential to resolve two enantiomers is very high given the variety of natural and derivatized cyclodextrins [5].

Choosing the type of cyclodextrin to be used is important as the α-, β-, and γ-cyclodextrins have different sized cavities and the mechanism of inclusion complexation necessitates that the enantiomers fit inside the cavities. The type of cyclodextrin is also important as different cyclodextrins have different solubilities and the use of certain cyclodextrins in certain solvents does not allow for the optimal concentration of cyclodextrin to be employed due to solubility limitations [12]. In some cases, derivatives of cyclodextrins offer solubilities greater than the solubilities of non-derivatized cyclodextrins. For example, hydroxypropyl and carboxymethyl groups increase the aqueous solubility limit of β-cyclodextrin from 1.8% (w/v) to 20% (w/v). This allows for a wider concentration range for separation optimization [3].

The degree of substitution and the position of the substituents for derivatized cyclodextrins also affect enantioseparations. For example, a single isomer sulfated cyclodextrin, which contains the same number of sulfate groups at the same positions on each glucose unit, was able to resolve the enantiomers of doxylamine while a mixed isomer sulfated cyclodextrin, which contains cyclodextrins with different numbers of sulfate groups at different positions on the glucose units, was unable to do so. In contrast, the mixed isomer sulfated cyclodextrin was able to resolve alprenolol while the single isomer sulfated cyclodextrin was unable to do so [16]. Furthermore, as moderate batch-to-batch reproducibility in separations is often observed for some derivatized cyclodextrins that are composed of a distribution of isomers, recent trends show a focus on single-isomer derivatives, which are more likely to yield reproducable separations [8].

The use of charged cyclodextrins expands the range of enantiomers that are able to be separated by uncharged cyclodextrins. As uncharged cyclodextrins can be used to separate charged enantiomers, charged cyclodextrins can be used to separate neutral enantiomers. In either case the charge on the enantiomers or cyclodextrin carries over to the enantiomer-cyclodextrin complexes [13]. It is important for the complexes to be charged as electrophoretic principles are governed by the charge densities, or charge-to-mass ratios, of analytes [9]. Neutral complexes can also be eluted, but the electroosmotic flow (EOF) would have to be strong enough to provide practical separation times [19]. Another option is the ion pairing of charged cyclodextrins with charged enantiomers. Ion pairing involves strong, attractive electrostatic interactions between cyclodextrins and enantiomers with opposite charges. Additionally, in order for the complexes resulting from ion pairing to have mobilities, the charges on the cyclodextrin and enantiomers should be unequal and opposite so that the complexes are charged [13]. Due to the strong electrostatic interactions, ion pairing is preferable to the use of uncharged cyclodextrins from the viewpoints of selectivity and speed [6]. Enantiomers and cyclodextrins that have the same net charge can make the separation ineffective due to the repulsive electrostatic forces between the enantiomers and cyclodextrin that would result [16].

//Cyclodextrin Concentration//

As previous discussed, cyclodextrin concentration is important in that there exists an optimal concentration of cyclodextrin for the resolution of enantiomers [5]. In addition, other effects are induced by altering the concentration of the cyclodextrin. One effect is an increase in the average size of the enantiomers as the concentration of cyclodextrin is increased. This results in longer migration times for the enantiomers as their charge-to-mass ratios are smaller [20]. Another effect of increasing cyclodextrin concentration, especially to concentrations above 20% (w/v), is an increase in viscosity, which also slows migration time [3].

//Ionic Strength of the Background Electrolyte (BGE)//

Increasing the ionic strength, or concentration, of the BGE, or buffer, reduces the zeta potential and hence reduces the EOF. This can result in longer migration times as the EOF is not driving the enantiomers toward the detector. The longer migration times can in turn result in greater resolution as the enantiomers and cyclodextrin would have more time to interact. Additional band broadening from the longer separation times would be minimal so that the principal disadvantage would be the longer separation times themselves. Furthermore, the maximum usable concentration of buffer is limited by the buffer’s conductivity so that increasing the buffer concentration would eventually increase the current to a point at which Joule heating significantly reduces separation efficiency [19].

Generally, triangular peaks in CZE are a consequence of a mismatch between the mobility of the BGE and that of the enantiomers [3]. The reason for the triangular shape of the peaks is electrodispersion, in which unbalanced electric fields at the boundaries of the zones containing the analytes create diffuse spreading of the zones [8]. The use of a buffer system with a mobility that better matches that of the enantiomers would correct for the problem of triangular peaks [3]. Specifically, triangular peaks exhibiting peak tailing can be a result of an ionic strength that is too low, which would cause a mobility mismatch between the BGE and enantiomers. As long as the mobilities of the BGE and enantiomers are compatible for certain concentrations of BGE, it is generally the case that a BGE concentration of at least a factor of 100 greater than the concentration of the enantiomers is sufficient to prevent peak distortion [20].

//pH//

The pH of the buffer plays an important role in determining selectivity and time of migration. The EOF in fused silica capillaries occurs at a pH higher than 3 due to the deprotonation of the silanol groups above this pH [15]. As the pH increases, the EOF increases, which decreases migration times and selectivities [20]. The pH of the buffer also controls the ionization state of enantiomers and cyclodextrins based on their pKa values. For example, carboxymethyl-β-cyclodextrin contains carboxyl groups that are all protonated at pH values below 4 and all deprotonated at pH values above 5. At pH values below 4, the cyclodextrin acts as a “quasi-stationary phase” as it is uncharged and only migrates as a result of the EOF. At pH values above 5, the cyclodextrin acts as a “moving stationary phase” in a micellar-like system due to the mobility of the cyclodextrin resulting from the negative charges, which are generated from deprotonation, on the cyclodextrin. At pH values below 4, carboxymethyl-β-cyclodextrin is neutral and can be used to separate charged enantiomers. At pH values above 5, the cyclodextrin is charged and so can be used to separate neutral enantiomers or enantiomers of the opposite charge via ion pairing [13].

Furthermore, choosing a pH that enables the optimization of a separation involves maximizing the buffer capacity. Maximum buffering capacity is obtained at a pH value that is equal to a pKa value of the buffer [15]. So, another consideration in choosing a buffer involves whether the buffer has a pKa value that is compatible with the pH of optimized separation conditions.

//Temperature//

Altering the temperature can result in contradictory effects that need to be balanced in enantioseparations. Heating can increase the solubility of the enantiomers and cyclodextrin, which would increase the probability of complexation and as a result increase the selectivity as long as the concentration of cyclodextrin is below the optimal concentration of cyclodextrin. However, heating can destabilize the complexes by disrupting hydrogen bonds and other weaker interactions [1]. Thus, for soluble enantiomers and cyclodextrins, decreasing the temperature tends to increase selectivity by stabilizing complex formation for when the concentration of the cyclodextrin is below the optimal concentration of the cyclodextrin [6]. Similarly, for soluble enantiomers and cyclodextrins, increasing the temperature can increase selectivity by weakening the stability of the complexes for when the concentration of the cyclodextrin is above the optimal concentration of the cyclodextrin [9]. The affect of temperature on solvent viscosity is also significant. Increasing the temperature decreases the solvent viscosity, which causes migration times and thus selectivity to decrease [20]. Another effect of altering temperature is the alteration of the pH of the BGE [14]. Overall, results from changes in temperature are usually much less pronounced than those from changes in pH, cyclodextrin concentration, type of cyclodextrin used, and ionic strength of the BGE [6].

//Polar Organic Solvents//

Adding modifiers, such as polar organic solvents, also provides a way to alter selectivity. Organic solvents typically compete with the enantiomers for the cyclodextrin, lowering the binding constants of the complexation reactions of the enantiomers with the cyclodextrin and increasing the optimum concentration of the cyclodextrin [6].

//Applied Voltage//

Effects from changing the applied voltage are not very different in enantioseparations involving cyclodextrins in CZE compared to other separations in CZE. In general, increasing the applied voltage increases the EOF, which shortens migration times, results in sharper peaks, and can improve resolution up to the point at which Ohm’s law no longer applies and Joule heating results [14].

//Multiple Cyclodextrins//

In some cases multiple cyclodextrins are used to resolve all enantiomers in mixtures of enantiomeric solutes in which each of the individual cyclodextrins is only able to resolve some of the enantiomeric solutes. Additionally, the use of multiple cyclodextrins is applied to cases in which one cyclodextrin permits resolution of enantiomers but does allow them to move past the detector or causes them to migrate very slowly. In these cases, the addition of a second cyclodextrin that moves in the direction of the detector and competes with the first cyclodextrin for the enantiomers has been shown to allow for faster separations or to allow for detection of the enantiomers in general [10]. Furthermore, applying a dual separation system, in which the buffer contains two cyclodextrins, typically improves resolution to an extent that is greater than the sum of the resolution improvements made by each cyclodextrin [8].

//Ligand Exchange Mechanisms//

It is also possible for cyclodextrins to take place in ligand exchange mechanisms that allow for the resolution of enantiomers in CZE. The enantioseparation of a racemic mixture of tryptophan utilizing a derivatized cyclodextrin and copper(II) was reported in which the enantioseparation was unable to take place without the use of copper(II). It was thought that the copper(II) ligated the derivatized cyclodextrin at two coordination sites and also ligated tryptophan at two coordination sites such that ligated L-tryptophan was able to be included in the cyclodextrin cavity while ligated and D-tryptophan was unable to be included in the cavity. As a result of this, the enantiomers of tryptophan had different binding constants with the cyclodextrin and were able to be resolved [11].


 * __Conclusions__**

Cyclodextrins offer a great amount of versatility for the separation of enantiomers in CZE. The large amount of derivatized cyclodextrins combined with the ability to alter parameters of CZE such as the ionic strength of the BGE, pH, temperature, and applied voltage allows for the separation of a variety of enantiomers. The ability to add polar organic solvents, more than one cyclodextrin, or metals, such as with ligand exchange mechanisms, to the separation scheme further increases the versatility of cyclodextrins in enantioseparations. Furthermore, the ability to predict the effective mobilities of the enantiomers and the optimum concentration of cyclodextrin are useful in expediting the optimization of separations as long as the required information regarding mobilities and binding constants is known. If the necessary information regarding mobilities and binding constants is unknown, certain estimations must be made to infer the appropriate choice of cyclodextrin and appropriate concentration of the cyclodextrin for the separation of enantiomers. These estimations are based on the structures of the cyclodextrins and enantiomers in the environment and at the pH of the BGE and separations of similar enantiomers involving cyclodextrins. As there is neither a universal set of rules for selecting an appropriate cyclodextrin for the separation of enantiomers nor comprehensive theories that describe the interactions of cyclodextrins with chiral compounds, the optimization of a separation of enantiomers involving cyclodextrins can be a tedious task in which various cyclodextrins, concentrations of cyclodextrins, and settings of other parameters may have to be tried. Further work that could be useful to performing enantioseparations with cyclodextrins includes studies to elucidate a universal set of rules for selecting an appropriate cyclodextrin and studies to more fully understand the inclusion complexation of cyclodextrins with chiral compounds.


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