On the Crystallization of Epitaxial Racemic Conglomerates - The

J. Phys. Chem. C 2010, 114, 21593–21604

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On the Crystallization of Epitaxial Racemic Conglomerates W. J. P. van Enckevort* Radboud UniVersity Nijmegen, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands ReceiVed: September 7, 2010

The resolution of racemic conglomerates by crystallization methods is often prevented by the formation of epitaxial racemic conglomerates. These laminar crystals composed of alternating layers of opposite enantiomers are formed by the interplay between mass transport and surface kinetics involving normal and epitaxial twodimensional nucleation. Three classes of epitaxial racemic conglomerates (ERC’s) are distinguished from crystallographic symmetry considerations. For the two most commonly occurring classes an analytical model has been developed that describes the “competition” of the two opposite enantiomers in their transport toward the crystal surface and their incorporation in the crystal by 2D nucleation growth. From this model and considering a limiting, stationary case, the conditions for ERC formation as a function of, among others, diffusion boundary layer thickness, supersaturation and epitaxial interfacial energy are derived. The dynamics of the oscillatory process of ERC growth is studied by numerical integration of the analytical expressions, using characteristic parameters for the solution growth of small molecule organic crystals. 1. Introduction Crystallization of racemic conglomerates is a powerful approach in the preparation of enantiomerically pure compounds, including pharmaceutical components. In this process the R and S enantiomers crystallize as separate solid phases from a supersaturated solution of the racemic compound. After growth, the mirror-image crystals can be sorted by hand as did Louis Pasteur1 in his study of a tartrate salt and more than a century later Sangwal et al.2 for potassium dichromate crystals. However, the direct resolution of conglomerates can be realized more efficiently by preferential crystallization through enantioselective seeding or by using suitable enantiomerically pure additives.3-5 Recently, abrasive grinding has been used to obtain a single chiral solid phase from an initially racemic mixture of conglomerate crystals in contact with a racemizing solution or an achiral solution.6-8 In this process the resolution outcome can be directed by using a minor initial excess of one enantiomer in the solid phase,7,8 a suitable enantiopure tailor-made additive,9 or circularly polarized light.10 The methods of manually sorting and enantioselective seeding of a supersaturated solution do not work if mutual epitaxial growth of the enantiomers occurs. The formation of epitaxial racemic conglomerates also poses a problem if large crystals of single handedness are required. This polyepitaxy phenomenon occurs if during crystal growth one enantiomer grows on the surface of the opposite one in an oriented manner. If this takes place repeatedly during crystallization, then a laminar structure of alternating homochiral layers develops, making the crystal racemic as a whole.11-14 The laminar structure of the crystals can be revealed by partial dissolution of one enantiomer in a solution saturated with the opposite enantiomer.11-13 These etching experiments showed that this layering is to some extent periodic, which indicates an oscillatory process in which the alternating layers form.11 * To whom correspondence should be addressed. Phone: +31 24 3653433. Fax: +31 24 3653067. E-mail: [email protected].

Figure 1. Oscillatory process of ERC formation: (i) growth of R enantiomer on R surface induces depletion of R solute, (ii) nucleation of S layer on R surface, (iii) growth of S on S surface induces depletion of S and accumulation of R solute, and (iv) nucleation of R layer on S surface, followed by growth of R on R surface in (i).

The growth mechanism leading to the formation of the laminar structure of alternating R and S layers in the epitaxial racemic conglomerate crystals has been clarified by Gervais et al.11 As shown in Figure 1, it involves interplay between volume diffusion of the R and S compounds toward the crystal surface and the two-dimensional nucleation of new R or S layers on the crystal surface. Starting from an R crystal surface, the R growth units adjacent to this surface are incorporated into the crystal. After some period of growth the concentration of R molecules near the surface gets depleted and the 2D nucleation of new R layers is strongly retarded. In that period the concentration of the S molecules remains constant and at a given moment the epitaxial nucleation of a new S layer on the R crystal surface becomes more favorable than the homogeneous nucleation of R on R. Now the growth of S layers sets on. In this period the concentration of S near the surface decreases and that of R increases as no R is consumed by the crystal. This process proceeds until R is preferentially nucleated on the S crystal surface and the process repeats from the beginning. This oscillatory process leads to the formation of the laminar structure of alternating homochiral layers. Evidence for this explanation was found by application of sufficient stirring, promoting mass transport of the desired enantiomer toward the crystal surface or by using a lower supersaturation, increasing

10.1021/jp108527h  2010 American Chemical Society Published on Web 11/12/2010

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J. Phys. Chem. C, Vol. 114, No. 49, 2010

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TABLE 1: Chiral Point Groups and Point-Group Symmetry of the Related Bravais Lattices chiral point group 1 2 222 3 32 4

point group related Bravais lattice 1j 2/m mmm 6mm 6mm 4mm

chiral point group

point group related Bravais lattice

422 6 622 23 432

4mm 6mm 6mm m3jm m3jm

the difference in 2D nucleation barrier of the opposite enantiomers.11,13 Both cases yielded enantiomerically pure crystals as under these conditions epitaxial nucleation of the opposite enantiomer is highly unfavorable. This paper elaborates on the phenomenon of epitaxial racemic conglomerate (ERC) development from a theoretical point of view. First, we describe the different classes of ERC in relation to the symmetry of the crystal lattice. Then, an analytical model for the formation of the ERC’s is developed, which describes the “competition” of the two opposite enantiomers in their transport toward the crystal surface and their incorporation into the crystal by two-dimensional nucleation. From this model and considering stationary case we derive the condition for the occurrence of ERC as a function of several parameters, such as boundary layer thickness, supersaturation, solute diffusion constant and epitaxial interface energy. Finally, we look at the dynamics of the process of oscillating crystallization of the opposite enantiomers by numerical integration of the analytical equations derived. 2. Epitaxy of Enantiomers and Lattice Symmetry The point group of a homochiral crystal has symmetry elements no other than rotation axes, including the 1 “axis”. Table 1 gives the Hermann-Mauguin symbols of these chiral point groups as well as the point group symmetry of the Bravais lattices of the related space groups. The introduction of an epitaxial crystal fragment of the opposite enantiomer in contact with a chiral crystal requires inversion or mirroring of this part with respect to the host crystal. This can be realized in several ways, depending on the extra symmetry operator used and the lattice point group symmetry of the crystals. On the basis of this we can distinguish three different classes of enantiomeric epitaxy. Class (i): Matching 3D Lattices. If the operator relating the crystal volumes of opposite enantiomer is also a symmetry operator of the crystal lattice point group, then the lattices of the different crystal parts “match”, i.e., have identical orientation. Apart from a possible slight translation less than a unit lattice vector length we have a perfect lattice match at the interface, which is often favorable from an energy point of view. By using standard single crystal X-ray diffraction techniques the ERC crystal appears as a single crystal as the reciprocal lattices of the enantiomeric parts coincide. If the growth rates of the opposite, polar faces of the homochiral crystal fragments are similar, then also the morphology of such a crystal appears as single. Because all Bravais lattices are centrosymmetric, all ERC’s of which the R and S fragments are related by simple inversion belong to this class. An example for point group 1 is given in Figure 2a. Upon using a mirror plane for linking the opposite crystal fragments in an ERC of this class, the mirror plane must be parallel to a mirror plane of the associated lattice point group. For instance, Figure 2b shows two opposite fragments of a chiral crystal with point group 222 which are

Figure 2. Examples of the three different ERC classes. (a) Class (i), crystal fragments related by inversion in point group 1. (b) Class (i), crystal fragments related by a mirror plane parallel to a-b plane in point group 222. (c) Class (ii), crystal fragments related by a mirror plane parallel to a-b plane in point group 2. (d) Class (iii), crystal fragments related by mirroring followed by an arbitrary (here 90°) rotation.

related by a mirror plane that coincides with the contact plane and is parallel to the a and b axes of the crystal. As this mirror plane is parallel to a mirror plane of the orthorhombic lattice point group mmm, the two lattices match. This situation was encountered for the ERC of Venlafaxine reported by van Eupen et al.12 Closer examination of the mirror related fragments in this group of ERC’s shows that these parts in fact are inverted with respect to each other, as in all the point groups of Bravais lattices with mirror planes, these planes are accompanied by perpendicular 2-fold axes. Here it should be realized that inversion can also be seen as mirroring followed by a 2-fold rotation about a perpendicular axis. So, class (i) of enantiomeric epitaxy just comprises the inversion of the two crystal fragments. Class (ii): Matching 2D Lattices at the Interface. If the mirror plane relating the two opposite enantiomer crystal volumes coincides with the contact plane, then the 2D lattices of both parts show (again apart from a possible minor translation) a perfect epitaxial lattice match at the interface. Here it has to be realized that this mirror plane is not allowed to be parallel to a mirror plane in crystal lattice point group. That case comes under class (i). An example is a mirror plane parallel to the a-b plane in point group 2, which is not a symmetry element of the monoclinic lattice point group 2/m (Figure 2c). By using X-ray diffraction the crystal appears as a “twin” showing the two opposite reciprocal lattices, which are mirrored along the reciprocal lattice plane parallel to the mirror plane in direct space. Although for this class it is expected that the mirror plane parallel to the contact plane generally coincides with a low index lattice plane, this is not a necessary requirement. Class (iii): No Matching 2D or 3D Lattices. If the inversion or mirror operation parallel to the interface is followed by an arbitrary rotation, not being a part of the lattice point group, then there is no match of the 3D lattices and no 2D match at the interface. An example is shown in Figure 2d. This also holds if the mirror plane relating the opposite enantiomer crystal volumes is neither parallel to the interface nor a part of the lattice point group. In general, this situation of poor match leads to high interfacial energy and thus is not likely to occur. It might happen in cases of pseudo symmetry or if the 2D lattices of two different faces (hkl) show a close match. It is expected that the classes (i) and (ii) ERC’s are the most commonly occurring ones, because of the perfect interfacial lattice match in both cases. This often lowers the interfacial energy between the crystal parts and thus reduces the barrier for epitaxial nucleation of the opposite enantiomer.

Crystallization of Racemic Conglomerates

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Figure 3. Mass transport followed by crystal growth: Schematic view of solution concentration profile, c(z,t), adjacent to crystal surface.

Crystals generally grow via the slowly growing F-faces.15,16 As epitaxial crystal growth sets on by formation of 2D nuclei on a growth face, it is expected that the contact face between the enantiomers corresponds with such an F-face. For the classes (i) and (ii) the slice energy (i.e., the bonds sticking sideways out of the growth layer15,16) is identical for the formation of homoepitaxial (R on R or S on S) nuclei and heteroepitaxial nuclei (R on S or S on R). This also holds for a special class iii) situation in which the opposite enantiomers are related to each other by a mirror plane perpendicular to the contact face. For these most common cases it is expected that the average step free energy of the different 2D nuclei does not differ much. It is the interfacial energy between the layers of different enantiomers that makes heteroepitaxial nucleation more difficult. 3. Model for Epitaxial Racemic Conglomerate Growth 3.1. Interplay Mass Transport and Crystal Growth. Crystal growth can be considered as a series coupling of volume diffusion of growth units toward the crystal surface followed by their incorporation into the crystal lattice. Transport proceeds through a diffusion boundary layer adjacent to the crystal surface, following Fick’s second law17

∂c(z, t) ∂2c(z, t) )D ∂t ∂t

(1)

as shown in Figure 3. In this equation c(z,t) is the concentration of R or S enantiomer molecules in the solution, z is the distance from the crystal surface and t is time. D is the solute diffusion coefficient, which is identical for the R and S enantiomer in an achiral solvent. The solution beyond the boundary layer of thickness, δ, is racemic with a concentration cbulk for each enantiomer: R,S cR(z g δ) ) cS(z g δ) ) cbulk

(2)

If the growing crystal surface is R-type, only R-solute will be incorporated into the crystal. Conservation of mass requires (Fick’s first law17)

RRR(cR(z ) 0, t)) ) -JR(z ) 0, t)Ω ) D

R

∂c (z, t) ∂z

|

z)0

RSS(cS(z ) 0, t)) ) -JS(z ) 0, t)Ω ) D

∂cS(z, t) ∂z

|

z)0

Ω

(4a)

and

RRS(t) ) 0

(4b)

Implementing these boundary conditions in Fick’s second law (eq 1) leads to a decrease of S solute concentration and an increase of R concentration at the growing S surface. This goes on until the rate of formation of an R layer on the S surface, RRS(cR(z ) 0,t), exceeds RSS(cS(z ) 0,t). Repeating this process leads to the sequential growth of the opposite enantiomers in the ERC. If the functions RSS(cS), RSR(cS), RRR(cR), and RRS(cR) are known, then the conditions for the presence or absence of ERC’s can be derived. If oscillatory growth takes place, then the time dependent growth rates and R, S solute concentration profiles in the boundary layer as a function of time as well as the width of the homochiral lamella in the crystals can be evaluated by numerical integration of Fick’s second law, using the boundary conditions 2-4. The numerical integration in time is realized by the finite difference method using the MatLab program EPICON given in the Supporting Information. 3.2. Homoepitaxial versus Heteroepitaxial 2D Nucleation Growth of Enantiomer Layers. The description of the growth process of ERC’s needs analytical expressions for the various growth rates, Rij(ci), with i,j ) R or S. We here use a model for homoepitaxial (i ) j) and heteroepitaxial (i * j) 2D nucleation growth by the Birth and Spread mechanism.18,19 The change in free energy upon the formation of a homo or heterepitaxial, circular 2D nucleus with radius r (figure 4a) on a crystal surface is given by

∆G(r) ) -

πr2hst πr2σij ∆µ + 2πrγst + Ω s

(5)

Ω

(3a)

and

RSR(t) ) 0

enantiomer on an R face respectively. It is obvious that a similar definition can be given for RSS(cS) ans RRS(cR). For a given S, R solute concentration, RRR > RSR and RSS > RSR. Continued growth of R layers leads to a depletion of R enantiomer near the crystal surface. This lowers cR(z ) 0,t) and thus results in a decrease of RRR growth rate. On the other hand, the concentration of the S enantiomer cS(z ) 0,t) does not change S . At a given moment the and remains its bulk value, cbulk SR S formation (growth) rate R (cbulk) of a an epitaxial S layer on top of the R crystal surface can exceed RRR. Then, a layer of S is formed instead of R, on top of which S crystallizes easily with rate RSS. The conservation of mass now requires

(3b)

Here JR denotes the flux of R growth units, each with volume Ω toward the crystal surface. RRR(cR) and RSR(cS) are the growth rates of R enantiomer on an R crystal surface and of S

In the first term, which comprises the energy change upon solidification, ∆µ ) µf - µs is the difference in chemical potential of a growth unit in solution and the bulk solid phase, hst is the step height, and Ω is the volume of one growth unit. The second term gives the edge free energy of the nucleus. As mentioned before, we assume that the step free energy, γst, is independent of the type of nucleus formed. The third term covers the change in interfacial and surface energy upon formation of the nucleus. In this part of expression (5) the change in interfacial plus surface energy per surface molecule area, s ) Ω/hst, is given by

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( )

πIVst2 R ) 1.14hst 3

1/3

(9)

, with I the formation frequency of supercritical nuclei per unit surface area and Vst the propagation velocity of a step. The step velocity is given by

[ (

Vst ) β exp

) ]

∆µ - σij -1 kT

(10)

with β a kinetic constant.18 The nucleation frequency is given by

I ) νZC(r*)

Figure 4. Schematics of two-dimensional nucleation in ERC growth. (a) Circular homo or heteroepitaxial 2D nucleus on crystal surface; (b) cross-section of R-type nucleus on S substrate; (c) cross-section of S-type nucleus on R-substrate.

σij ) 0

if

i)j

(homoepitaxial nucleation)

(6a)

with ν the frequency of attachment of one growth unit to a critical nucleus, Z the Zeldovich correction term, and C(r*) the concentration of critical 2D nuclei on the surface.18 The attachment frequency, ν, is given by the circumference of a critical nucleus times the probability of attachment of a growth unit per unit time per unit step length, Γcs/s1/2, with Γ a kinetic constant and cs ) ceq exp(∆µ/kT) the R or S solute concentration adjacent to the surface

and

ν ) 2πr* σij ) σRS

or

σSR

Γceq s1/2

exp(∆µ/kT)

(12)

(heteroepitaxial nucleation)

(6b) R RS S + σint - σsurf for the formation of an R-type Here σRS ) σsurf S SR + σint nucleus on an S substrate (Figure 4b) and σSR ) σsurf R σsurf for an S nucleus on an R substrate (Figure 4c). In these i expressions σsurf is the surface free energy of an enantiomer i ij the interfacial energy of an i layer on crystal surface and σint top of a j substrate. For classes (i) and (ii) ERC’s σRS * σSR, R S RS SR because σsurf * σsurf and σint * σint (Figure 4). Only in the case of a class iii) ERC with a mirror plane perpendicular to the contact plane, σRS ) σSR. To simplify our model we approximate σRS ≈ σSR in the following. Solving ∂∆G(r)/∂r ) 0 for r gives the radius of a critical 2D nucleus

r*ij )

(11)

γsts [∆µ - σij]

(7)

C(r*) ) C0 exp(-∆G*/kT)

πγst2s (∆µ - σij)

(8)

A first conclusion that can now be drawn is that if ∆µ decreases to σij, ∆G*ij approaches infinity and no nucleation of R on S (or reversed) will occur. This implies that for ∆µ < σij (i * j) no ERC can be formed. The crystal growth rate by the Birth and Spread 2D nucleation mechanism is given by18

(13)

with C0 the number of growth sites per unit crystal surface area, which is equal to 1/s. Finally, the Zeldovich factor to correct for the deviation from the ideal Boltzmann distribution of 2D nuclei is given by

Z)

Q ( 2πkT )

1/2

(14a)

with

Q)-

and a nucleation barrier

∆G*ij )

The kinetic constant Γ is the frequency of addition of a growth unit at a single step site from a solution containing one concentration unit of solute. It is related to the kinetic constant for step propagation according to β ) Γs1/2ceq. The concentration of 2D nuclei equals

(

∂2∆G(N) ∂N2

)

(14b) N)N*

Using N ) πr2/s (N* ) πr*2/s), which is the number of growth units in a (critical) nucleus, the Zeldovich factor is elaborated to be

Z)

(∆µ - σij)3/2 1 2πγst skT

1/2

( )

(15)

Substitution of eqs 12, 13, and 15 into eq 11 gives a nucleation frequency


(

I ) ΓceqC0

∆µ - σij kT

)

1/2


D Ji ) - (cbulk - csi ) δ

(∆µ - σij)1/2 exp(∆µ/kT) × exp(-∆G*/kT) (16)

From the above, the growth rate via homogeneous and heterogeneous 2D nucleation can now readily be calculated by substituting the expressions for I and νst into the Birth and Spread growth rate eq 9

(

Rij ) Akin

∆µ - σij kT

) ( )[ ( 1/6

exp

) ] ( )

2/3 ∆µ - σij ∆µ exp -1 × 3kT kT ∆G*ij exp (17a) 3kT

with a kinetic factor Akin, which is independent of interface energies and supersaturation

Akin

(21)

(

πΓC0ceqβ2s1/2 ) 1.14hst 3

)

1/3

(17b)

As Ji ) -Rii(csi )/Ω and using eq 17a for Rii with σii ) 0, we obtain

[

( )] ( ) ( ) [ ( ) ] ( )

∆µsi ∆µsi 1/6 ∆µsi ΩD cbulk - ceq exp ) Akin exp × δ kT kT 3kT 2/3 πγst2s ∆µsi exp - 1 exp (22) kT 3∆µi kT s

with ∆µsi /kT the supersaturation of enantiomer i at the crystal surface. For an ERC to be formed the rate of nucleation of the opposite enantiomer

Rji ) Akin

( ) ( ∆µ# kT

1/6

exp

Using C0 ) 1/s and β ) Γs ceq, we come to 1/2

Akin ) 1.14hstΓceq

π 3

1/3

()

(18)

If Rii > Rji, then homogeneous nucleation of enantiomer i will continue; if Rji > Rii then an epitaxial layer of the opposite enantiomer j will be formed. 4. Criterion for the Formation of Epitaxial Racemic Conglomerates Placing a solid enantiomer i in a homogeneous, supersaturated racemic solution leads to a depletion of solute i in the vicinity of the growing crystal surface. The concentration profile of the j opposite enantiomer j, however remains unchanged at cbulk . If i the surface concentration, cs, of i becomes thus low that Rji(cbulk) > Rii(csi ), then layers of the opposite enantiomer will grow, leading to an ERC. However, if at t f ∞ the growth rate Rii(csi ) still exceeds Rji(cbulk) , then a stationary situation is obtained, and an enantiopure crystal instead of an ERC will be formed. For this stationary situation Fick’s second law becomes

(

)

(23)

with ∆µ# ) ∆µsj ) ∆µbulk - σji must exceed the right or lefthand term of eq 22. If we consider Rji ) Rii as the transition criterion for the occurrence or absence of ERC formation and we neglect the difference σji between ∆µbulk and ∆µ# in the factor exp(∆µbulk/3kT) of eq 23, then we can put the right-hand terms of eqs 22 and 23 equal. This implies ∆µsi = ∆µ#. Substituting this result in eq 22 our criterion for ERC formation now becomes explicit

[

( )] ( ) ( ) [ ( ) ] ( )

ceqΩD cbulk ∆µ# - exp δ ceq kT exp

) Akin

∆µ# -1 kT

∆µ# kT

2/3

1/6

exp

exp -

∆µ# × 3kT

πγst2s

3∆µ#kT

(24)

Rewriting this equation gives

[

B

( )] ( ) ( ) [ ( ) ] ( ( ))

cbulk ∆µ# - exp ceq kT

)

∆µ# kT

exp

2 i

∂ c (z) )0 ∂z2

)[ ( ) ]

2/3 ∆µbulk ∆µ# exp -1 × 3kT kT πγst2s exp 3∆µ#kT

(19)

1/6

exp

∆µ# -1 kT

∆µ# × 3kT 2/3

exp -E

∆µ# kT

-1

with with boundary conditions (i) ci(z g δ) ) cbulk (ii) Rii(csi )/Ω ) -Ji(z ) 0) ) D(∂ci(z)/∂z)|z)0 (iii) ci(z ) 0) ) csi with Ji the diffusion flux of growth units toward the crystal surface. Solving this problem gives the linear concentration profile for z e δ

cbulk - csi ci(z) ) z + csi δ and a diffusion flux

(20)

ΩD ceq B) , δ Akin

E)

πγst2s

, 3(kT)2

and

( )

cbulk ∆µ# ) ln kT ceq σij (25) kT

In the above expression we have four dimensionless variables B, E, cbulk/ceq, and σij/kT. The experimentalist can only vary B (by changing the diffusion boundary layer thickness, δ) and cbulk/ ceq (control of supersaturation). Expressing B in terms of the other variables gives a suitable (approximate) criterion for the occurrence of ERC’s

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B< ∆µ# kT


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( ) [ ( )][ ( ) ] [ ( ) ] ( )] [ 1/6

1 ∆µ# exp 3 kT

∆µ# exp -1 kT cbulk ∆µ# - exp ceq kT

2/3

∆µ# exp -E kT

-1

(26)

If the solute concentrations are expressed in terms of kmol/m3 instead of the number of growth units per m3, then Ω in the expression for B must be replaced by Vgu ) 103NAvΩ, with NAv being Avogrado’s number. Typical value ranges of D, δ, ceq, γst, s, σij, Γ, Akin, and ∆µ/ kT and the related values of E and B are summarized in Table 2 for smaller organic molecules. The value ranges of Γ and Akin are estimated by using the relations A1 and 18 in the Appendix. In Figure 5, B values according to eq 26 are plotted as a function of the applied supersaturation ∆µ/kT ) ln(cbulk/ ceq) for different values of σij/kT and E. In the area left and above the transition line B(∆µ/kT), i.e., at high B values (small

TABLE 3: Parameters Used for Example of ERC Formation D γst σij/s Akin s

10-9 m2/s 7.5 × 10-12 J/m 5 mJ/m2 2.5 × 10-6 m/s 0.25 × 10-18 m2

T ∆µ/kT (as input) E σij/kT ceq

300 K 0-1.0 0.86 0.28 0.5 × 103 mol/m3

boundary layer thickness, δ) and low supersaturation, no ERC’s are formed for the given σij/kT and E values. As an example, which will also be elaborated in the section on the dynamics of ERC formation, we take some arbitrary, but realistic parameters for the growth of an organic crystal. These are summarized in Table 3. Figure 6 gives values of B that correspond with the transition line between single crystal growth and ERC formation as a function of the driving force ∆µ/kT. The same figure also displays the related boundary layer thickness, δ, and the crystal growth rate, R, which are derived by using eqs 25 and 24 respectively. It can readily be recognized from Figure 6b that for ∆µ/kT less than ∼0.45 nonrealistic boundary layer thicknesses are required to obtain ERC growth.

TABLE 2: Parameter Ranges for the Solution Growth of Crystals of Smaller Organic Molecules D δa γst ≈ σsurfs1/2 b σij/s Γ s (≈hst2 ≈ Ω2/3)

10-9 m2/s 10-5-10-3 m 2.5-25 × 10-12 J/m 1-30 mJ/m2 typically 103 m3kmol-1 s-1 0.25 × 10-18 m2

c

ceq T Akin ∆µ/kT ) ln(cbulk/ceq) E B

0.1-10 kmol/m3 ∼300 K typically 10-6 m/sc 10-2-2 0.05-2.0 10-3-10

a Under zero gravity conditions without stirring or for growth from a gel, δ approaches infinity. b Estimated from solid-liquid surface energies ranging from 5-50 mJ/m2.20,21 c But can be one or a few orders of magnitude larger or 1 order of magnitude smaller.

Figure 5. Transition lines B(∆µ/kT) for the occurrence of ERC growth for different E and σij values. In the area left and above the transition lines (i.e., at higher B values and lower supersaturation, ∆µ/kT) no ERC’s are formed. (a) E ) 0.15, (b) 0.7, (c) 3.0.



Figure 7. Dynamics of ERC formation using the parameter set of Table 3, ∆µ/kT ) 0.7 and δ ) 150 µm. (a) Growth rate of the R and S enantiomers as a function of time. (b) Interface concentration of both enantiomers as a function of time. (c) Lamellar structure of the ERC crystal obtained.

Figure 6. Transition lines B(∆µ/kT) (a), δ(∆µ/kT) (b), and R(∆µ/kT) (c) for the occurrence of ERC growth using the parameter set of Table 3. In the areas left and above the transition lines only enantiopure crystals form. The dot in each graph indicates the transition point for the conditions used in the dynamic calculations of section 5.2.

At ∆µ/kT ) σij/kT ) 0.28, δ goes to infinite, which implies that for lower supersaturation no ERC can be formed at all, even under zero gravity conditions or for gel growth. The MatLab program TransEpiCon used is given in Supporting Information. To demonstrate the dynamics of ERC formation in paragraph 5, we select ∆µ/kT ) 0.7, which corresponds to B ) 0.115, δ ) 1.13 × 10-4 m, and a growth rate of R ) 1.76 × 10-7 m/s as indicated in Figure 6. 5. Dynamics of ERC formation 5.1. General Features. The development of an ERC is evaluated by numerical integration of differential eq 1 in time for the two opposite enantiomers by using a finite difference

method (programm EPICON, Supporting Information). In this calculation the boundary conditions eqs 2, 3a, and 4a are imposed, of which the last two are used alternately. Expression (17) is substituted for the growth rates Rij in eqs 3b and 4b. Figure 7 shows the time dependent growth rate and surface solute concentration as well as the segmented ERC structure of the grown crystal for the conditions given in Table 3, ∆µ/kT ) 0.7 and a boundary layer thickness δ ) 150 µm. It can clearly be seen that the growth rates of the R and S enantiomers oscillate in time (Figure 7a). After an initial rapid decrease in growth velocity of enantiomer R followed by a longer period of slow decrease, the growth of R stops and is taken over by S, following the same sequence. This oscillation of growth rate is induced by the periodic change in solute surface concentration due to mass transport limitation (Figure 7b). During growth of R the interface concentration of R decreases; if S takes growth over, the interfacial concentration of R rises again as no R is consumed at the interface. The same holds for S. The resulting widths of the alternating R and S lamellas in the crystal is 3.11 µm (Figure 7c). For frequent domain changes, which occur for high supersaturation and low σij values, the growth rate, surface solute concentrations, and the domain width decrease in time until a stationary situation is attained. This is shown in figure 8 for the standard conditions and using δ ) 250 µm and ∆µ/kT ) 1.0. In the measurements that follow, we only consider the steady state values of the frequency of domain alternation, average growth rate and R/S domain spacing. 5.2. Dependence on Boundary Layer Thickness. A first parameter that can be controlled by the experimentalist is the boundary layer thickness. Figure 9 displays the frequency of R/S domain formation, the R/S domain width, and the average growth rate of ERC crystals as a function of δ. The parameters in Table 3 are used, keeping ∆µ/kT ) 0.7. Figure 9a shows that the frequency of R/S domain formation steadily increases for decreasing δ up to a maximum value of 0.068 s-1 at δ ) 120 µm and then, in a second δ regime, rapidly goes down to zero at δ ) 99.75 µm. For values lower than δ ) 99.75 µm

21600


Figure 8. Evolution toward a constant frequency of R/S domain formation in ERC growth. (a) Growth rate of R enantiomer as a function of time. (b) Lamellar structure of the ERC crystal obtained. In this simulation the parameter values of Table 3 are used with δ ) 250 µm and ∆µ/kT ) 1.0.

only enantiopure material crystallizes. This critical value for ERC formation is close to the value of δ ) 113 µm, which is derived by using eq 25, as discussed at the end of paragraph 4 and shown in Figure 6b. The R/S domain spacing shows a similar, but inverted, behavior as the frequency of domain formation (Figure 9b). For boundary layer thicknesses from 500 to 120 µm the spacing decreases linearly with δ from 7 to 3

Enckevort µm and then shows a sharp increase to (in theory) infinite at δ ) 99.75 µm. The average growth rate shows a gradual increase for decreasing boundary layer thickness as expected (Figure 9c). However, in the regime 120-99.75 µm, the growth rate shows a short decrease, after which it continues to increase. The growth rate at the second point of reversal is 1.94 × 10-7 m/s () 0.70 mm/h), which is not far from the value of R ) 1.76 × 10-7 m/s predicted in the previous paragraph. The two regimes of ERC formation can be qualitatively explained as follows. Figure 10, panels a and b, shows the evolution of the c(z,t) concentration profile and the associated growth rate Rii(t) during the formation of one R or S domain in the regime of larger δ. It is clear that the c(z,t) profile remains curved, i.e., d2c(z,t)/dz2 remains large, which according to eq 1 leads to a rapid decrease in interface concentration and thus in growth rate. If the boundary layer gets thinner, then the interface supersaturation reaches the value for which Rij > Rii sooner and the opposite enantiomer will start to grow. This leads to an increased frequency of domain formation and a decrease in domain width for decreasing δ. If δ approaches the critical value for enantiopure crystal growth, we enter the second regime of ERC formation, where the frequency rapidly decreases for decreasing δ values. From the evolution of the concentration profile, shown in Figure 10c, it can be seen that after a rapid decrease in c(z,t) curvature, this profile becomes more and more straight. As here d2c(z,t)/dz2 becomes very low, the change in interface concentration, dc(z ) 0,t)/dt, almost vanishes and a long period of almost constant interface concentration and thus growth rate follows (Figure 10d). As a consequence of this long period of constant growth rate, the frequency

Figure 9. ERC formation as a function of boundary layer thickness, δ. (a) Frequency of R/S domain formation. (b) Width of R/S domains. (c) Growth rate. The parameters of Table 3 are used keeping ∆µ/kT ) 0.7.



Figure 10. The two regimes of ERC growth. In the regime of larger boundary layer thickness the solute concentration profiles remain curved in time (a) and the growth rate still shows a substantial decrease at the end of each cycle of R or S formation (b). If the boundary layer thickness approaches the critical point of ERC formation, then the solute concentration profile evolves toward an almost straight line (c) and after an initial period of rapid decline, the growth rate remains virtually constant for a long period (d). Parameters for (a and b): Table 3, δ ) 500 µm and ∆µ/kT ) 1.0; for (c and d): Table 3, δ ) 25.80 µm and ∆µ/kT ) 1.0.

goes down and the R/S domain width increases for decreasing δ. At the critical δ (and below) the period of constant growth rate is infinite as the c(z,t f ∞) profile becomes an exactly straight line. The same behavior of the frequency of R/S domain formation, the R/S domain width and the average growth rate of ERC crystals as a function of δ was also found for other parameter sets, leading to R/S domain widths ranging from a few tens of nanometers to tens of micrometers. 5.3. Dependence on Supersaturation. The second parameter that can readily be managed is supersaturation, ∆µ/kT ) ln(cbulk/ceq). Figure 11 displays the frequency of R/S domain formation, the R/S domain width and the average growth rate of ERC crystals as a function of ∆µ/kT, using the parameter set of Table 3 and keeping δ ) 250 µm. This boundary layer thickness is typical for free convection around a crystal of several millimeters to centimeters in size in an unstirred solution at g ) 9.81 m/s2. Upon lowering supersaturation, the frequency of domain formation gradually decreases until ∆µ/kT = 0.61, followed by an sharp decrease (i.e., dfreq/d∆µ/kT f ∞, see inset Figure 11a) to zero at ∆µ/ kT ) 0.602. The R/S domain spacing shows a continuous increase for decreasing supersaturation up to a critical point ∆µ/kT ) 0.602, where the spacing becomes infinite and an enantiopure crystal is obtained (Figure 11b). As expected, the average growth rate decreases for decreasing supersaturation, showing a “kink” at the critical point (Figure 11c). For very low supersaturations (not shown here), the growth rate by 2D nucleation becomes extremely low and the spiral

mechanism will determine crystal growth, leading to enantiomerically pure crystals. 5.4. Dependence on “Interfacial” Energy, σij. Figure 12 gives the frequency of R/S domain formation, R/S domain width and the average growth rate of ERC crystals as a function of the “interfacial” energy, σij. In contrast to δ and ∆µ/kT, this parameter can not be controlled by the experimentalist and is determined by the crystal-solvent system. Again the parameters of Table 3 are used, keeping ∆µ/kT ) 1.0 and δ ) 100 µm. Both the frequency and domain width are largely dependent on σij. The frequency of domain formation goes down from 20 s-1 for σij ) 1 mJ/m2 to zero at the critical value σij ) 8.437 mJ/m2 (Figure 12a). For higher σij only enantiopure crystals are formed under these conditions. The large dependence of the frequency on σij has dramatic implications for the R/S domain width, which rises from 35 nm for σij ) 1 mJ/m2 to infinite at the critical value of σij (Figure 12b). For very low values of σij (i.e., < 1 mJ/ m2), the R/S spacing goes to one or a few lattice spacings (not shown here) and the ERC approaches a solid solution or a racemic crystal. Despite the fact that the supersaturation is kept constant, the average growth rate increases for decreasing σij (Figure 12c). This is readily explained by the fact that the frequency of domain swapping is larger for lower σij. In this way a large depletion of R or S solute adjacent to the crystal surface is prevented, keeping the average solute concentration and thus the average

21602


Enckevort

Figure 11. ERC formation as a function of supersaturation, ∆µ/kT. (a) Frequency of R/S domain formation; (b) width of R/S domains; (c) growth rate. The parameters of Table 3 are used keeping δ ) 250 µm.

Figure 12. ERC formation as a function of “interfacial” free energy, σij (J/m2). (a) Frequency of R/S domain formation; (b) width of R/S domains; (c) growth rate. The parameters of Table 3 are used keeping δ ) 100 µm and ∆µ/kT ) 1.0.

growth rate high. For σij f 0, there is no distinction between the R and S components and the growth rate becomes identical

to that of growth from an enantiopure solution, but with twice the concentration of R or S solute.



6. Discussion and Conclusions Three classes of epitaxial racemic conglomerates can be distinguished on the basis of crystallography. The first and second class are based on a matching of the 3D lattices and the 2D interfacial lattices of the opposite enantiomers respectively. These relations are absent for the third category. For the first two classes, which are the most common ones, a kinetic model has been developed for the occurrence and dynamics of ERC formation. A main conclusion that follows from the model and the examples based on it, is that undesired ERC formation can be avoided by reducing boundary layer thickness and supersaturation. The boundary layer thickness can be lowered by stirring the solution, as was demonstrated by Gervais et al. for 5-ethyl5-methylhydantoin crystals.11 Interesting is the fact that our dynamics calculations show that reducing the boundary layer thickness first results in a decrease of the R/S domain width up to a minimum value, which then rapidly increases to infinite if δ lowers further approaching the critical value for ERC formation. The second approach that can be used to suppress ERC formation is lowering supersaturation. A very powerful way in avoiding ERC formation and at the same time yielding an enantiopure batch of crystals is by grinding a racemic mixture of (ERC) crystals in contact with a solution in which racemization takes place, as shown by Kaptein et al.13 Here the supersaturation is extremely low as a consequence of the near equilibrium conditions in this process. In addition, because of the vigorous stirring and the very small crystal size the boundary layer is very thin. Both conditions largely hamper ERC formation. A problem in quantitatively predicting the occurrence or absence of ERC formation is the fact that a number of parameters involved, namely Akin, σij and γst are rarely known for a given system. In principle Akin and γst can be determined by in situ observation of step growth processes on a crystal surface using AFM22,23 or optical microscopy.24 An alternative approach is using molecular dynamics/mechanics calculations to evaluate the three parameters. In principle, the basic ideas of this study can also be used for the growth of lamellar crystals composed of two different compounds that are epitaxially related. To realize such structures one needs, apart from a low σij to facilitate epitaxy, a large δ and ∆µ/kT. A complication in modeling the growth of these lamellar crystals is a, generally, different step free energy for the two compounds, which is not included in our model. Appendix: Estimation of Γ and Akin In the main text it is shown that

Akin ) 1.14hstΓceq

( π3 )

1/3

(18)

with Γ being the frequency of addition of one growth unit per step site from a solution containing one unit concentration (e.g., 1 kmol/m3) of solute. For direct integration of the growth units from the solution at the kink sites of the steps, this factor is given by25

Γ=

(

fFk ∆GΞ exp τD kT

)

(A1)

Assuming similar sizes of the solvent and solute units, f is roughly the molar fraction of solute at one unit concentration. For 1 kmol/m3 and a molecule volume of 1.25 × 10-28 m3, f ) 0.075 m3/kmol. The factor Fk is the kink density, i.e., the average number of kinks per step site, which ranges from 0.25 to 0.75 for many solution grown organic crystals. ∆GΞ is the free enthalpy of activation for the incorporation of a growth unit into a kink site at the step. τD is the time needed for a growth unit to leave the solution layer in contact with the crystal surface. As the mean free path of a growth unit in the solution is given by 〈x〉 ) 2(Dt/π)1/2, with time t,26 and assuming that 〈x〉 ≈ s1/2, it follows that the time τD is approximately

τD )

πs 4D

(A2)

Using D ) 10-9 m2/s and s ) 0.25 × 10-18 gives τD ) 2.0 × 10-10 s. A rather unknown parameter is ∆GΞ, which depends on the system involved. An average value of 28 kJ/mol has been reported in literature for systems ranging from inorganic salts, organics to macromolecules.25 Using this value, T ) 300 K, the above estimated values for f and τD and taking Fk ) 0.5 we come to Γ ) 2.5 × 103 m3 kmol-1 s-1 and by using eq 18 and ceq ) 1 kmol/m3 Akin ) 1.5 × 10-6 m/s. As the precise value of ∆GΞ is rarely known and surface diffusion can play a role as well,25 Akin, can be one or a few orders of magnitude larger or smaller. The largest possible (but unlikely) value for Akin is for ∆GΞ ) 0, in which case step propagation is completely diffusion controlled. Then Γ ) 1.9 × 108 m3 mol-1 s-1 and Akin ) 1.1 × 10-1 m/s. Nomenclature Akin B c cbulk ceq cs C0 C(r*) D E g G, ∆G ∆G* hst I i j J k r r* R s t Vgu

kinetic factor for 2D nucleation growth (m/s) dimensionless parameter: B ) (ΩD/δ)(ceq/Akin) concentration of solute (molecules/m3 or kmol/m3) R S bulk concentration of each enantiomer: cbulk ) cbulk 3 3 (molecules/m or kmol/m ) R S equilibrium concentration ceq ) ceq (molecules/m3 or kmol/m3) concentration of solute adjacent to crystal surface: c(z ) 0,t) density of possible nucleation sites on crystal surface (m-2) equilibrium concentration of critical 2D nuclei (m-2) solute diffusion coefficient (m2/s) dimensionless parameter: E ) (πγst2s)/(3(kT)2) gravitational acceleration on earth free enthalpy (J) barrier for 2D nucleation step height (m) formation frequency of supercritical 2D nuclei per unit surface area (s-1 m-2) R or S R or S flux of growth units (molecules/m2 s) Boltzmann constant (J/K) radius 2D nucleus (m) radius critical 2D nucleus (m) crystal growth rate (m/s) surface area growth unit (m2) time (s) volume of one kmol growth units (m3/kmol)

21604 Vst z Z

J. Phys. Chem. C, Vol. 114, No. 49, 2010 step velocity (m/s) distance from crystal surface (m) Zeldovich correction term

Greek symbols β Γ γst δ ∆µ ∆µs ∆µbulk ∆µ# ν σ

σsurf σint Ω

kinetic constant for step propagation (m/s) kinetic constant for attachment of growth units at step (m3/s or m3 s-1 kmol-1) step free energy (J/m) boundary layer thickness (m) difference in chemical potential of growth unit in solution, µf, and in crystal, µs (J/molecule) driving force at crystal surface (J/molecule) total driving force (J/molecule) ∆µbulk - σij (i * j) (J/molecule) attachment frequency of growth units to critical 2D nucleus (s-1) change in interfacial, σint, and surface free energy, σsurf, per surface molecule area upon addition of a growth layer (J/m2) surface free energy (J/m2) interfacial free energy between layers of opposite enantiomer (J/m2) volume of one growth unit (m3)

Supporting Information Available: Two MatLab (The MathWorks Inc.) programs for calculation of the dynamics of ERC formation (EPICON: SI1) and the criteria of ERC formation (TransEpiCon: SI2). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Pasteur, L. C. R. Hebd. Seanc. Acad. Sci. Paris 1848, 26, 535. (2) Sangwal, K.; Szurgot, M.; Szczepaniak, M. J. Cryst. Growth 1986, 79, 185. (3) Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolution; Krieger: FL, 1994. (4) Addadi, L.; Berkovitch-Yellin, Z.; Domb, N.; Gati, E.; Lahav, M.; Leiserowitz, L. Nature 1982, 296, 21. (5) Coquerel, G. Top. Curr. Chem. 2007, 269, 1.

Enckevort (6) Viedma, C. Phys. ReV. Lett. 2005, 94, 065504. (7) Noorduin, W. L.; Izumi, T.; Millemaggi, A.; Leeman, M.; Meekes, H.; van Enckevort, W. J. P.; Kellogg, R. M.; Kaptein, B.; Vlieg, E.; Blackmond, D. G. J. Am. Chem. Soc. 2008, 130, 1158. (8) Noorduin, W. L.; Meekes, H.; van Enckevort, W. J. P.; Millemaggi, A.; Leeman, M.; Kaptein, B.; Kellogg, R. M.; Vlieg, E. Angew. Chem., Int. Ed. 2008, 47, 6445. (9) Noorduin, W. L.; Asdonk, P.; Meekes, H.; van Enckevort, W. J. P.; Kaptein, B.; Leeman, M.; Kellogg, R. M.; Vlieg, E. Angew. Chem. Int Ed. 2009, 48, 4581. (10) Noorduin, W. L.; Bode, A. A. C.; van der Meijden, M.; Meekes, H.; van Etteger, A. F.; van Enckevort, W. J. P.; Christianen, P. C. M.; Kaptein, B.; Kellogg, R. M.; Rasing, T.; Vlieg, E. Nature Chem. 2009, DOI: 10.1038/NCHEM.416. (11) Gervais, C.; Beilles, S.; Cardinae¨l, P.; Petit, S.; Coquerel, G. J. Phys. Chem. B 2002, 106, 646. (12) van Eupen, J.; Th, H.; Elffrink, W. W. J.; Keltjes, R.; Bennema, P.; de Gelder, R.; Smits, J. M. M.; van Eck, E. R. H.; Kentgens, A. P. M.; Deij, M. A.; Meekes, H.; Vlieg, E. Cryst. Growth Des. 2008, 8, 71. (13) Kaptein, B.; Noorduin, W. L.; Meekes, H.; van Enckevort, W. J. P.; Kellogg, R. M.; Vlieg, E. Angew. Chem. 2008, 47, 7226. (14) Potter, G. A.; Garcia, C.; McCague, R.; Adger, B.; Collet, A. Angew. Chem. 1996, 108, 1780. Angew. Chem., Int. Ed. Engl. 1996, 35, 1666. (15) Hartman, P.; Perdok, W. G. Acta Crystallogr. 1955, 8 (49), 521 and 525. (16) Bennema, P. Handbook of Crystal Growth; Hurle, D. T. J., Ed.; North-Holland Elsevier: Amsterdam, The Netherlands, 1993; Vol. 1, Chapter 7, p 477. (17) Incropera, F. P.; Dewitt, D. P.; Bergman, T. L.; Lavine, A. S. Fundamentals of heat and mass transfer, 6th ed.; John Wiley and Sons: New York, 2006. (18) van de Eerden, J. P. Handbook of Crystal Growth; Hurle, D. T. J., Ed.; North-Holland Elsevier: Amsterdam, The Netherlands, 1993; Vol. 1, Chapter 6, p 307. (19) Kaschiev, D. Nucleation: Basic Theory with Applications; Butterworth-Heineman: Oxford, U.K., 2000. (20) Lu, H. M.; Wen, Z.; Jiang, Q. J. Phys. Org. Chem. 2007, 20, 236. (21) Winn, D.; Doherty, M. F. AIChE J. 1998, 44, 2501. (22) Heijna, M. C. R.; van den Dungen, P. B. P.; van Enckevort, W. J. P.; Vlieg, E. Cryst. Growth Des. 2006, 6, 1206. (23) Chernov, A.; Rashkovich, L.; Gvozdev, N. J. Phys.: Condens. Matter 1999, 11, 9969. (24) See for instance: Sunagawa, I.; Tsukamoto, K.; Maiwa, K.; Onuma, K. Prog. Cryst. Growth Charact. 1995, 30, 153–190. (25) Chen, K.; Vekilov, P. G. Phys. ReV. E 2002, 66, 021606. (26) Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; Oxford University Press: Oxford, U.K., 2002; p 852.

JP108527H

On the Crystallization of Epitaxial Racemic Conglomerates - The

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