Needle-Shaped Crystals - American Chemical Society

Article pubs.acs.org/crystal

Needle-Shaped Crystals: Causality and Solvent Selection Guidance Based on Periodic Bond Chains Michael A. Lovette† and Michael F. Doherty*,‡ †

Small Molecule Design and Development, Eli Lilly and Company, Indianapolis, Indiana 46285, United States Department of Chemical Engineering, University of California Santa Barbara, Santa Barbara, California 93106-5080, United States

‡

S Supporting Information *

ABSTRACT: Needle-shaped crystals, typified by aspect ratios of (1:1:100−1000) are often the steady-state growth shapes in the crystallization of active pharmaceutical ingredients from solution. Crystals with such high aspect ratio shapes are troublesome in the subsequent processing steps required in the formation of a drug product. Therefore, the ability to design crystallizations that directly avoid the formation of needles is of significant interest. In this article, a causality for the formation of needles and guidance for solvent selection based on this causality are provided. The causality presented in this article was formed based on a spiral growth model and requires the presence of a single strongest periodic bond chain within the lattice that is parallel to the direction of elongation of the crystal. The article provides a method for predicting the formation of a needle for a given solute−solvent system based on this causality. Furthermore, three case studies are provided that demonstrate good agreement between experimentally obtained and predicted shapes. As a result, generalized guidance for solvent selection based on the objective of avoiding needles is provided.

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INTRODUCTION Needle-shaped crystals (hereafter referred to as needles), as typified by aspect ratios in the range of (≈1:1:100−1000), are often formed in the crystallization of active pharmaceutical ingredients (APIs) from solution. This shape can especially be troublesome in the subsequent processing (e.g., filtering, drying, blending, tabletting) steps required in the formation of a drug product.1,2 Therefore, the ability to design crystallization processes to avoid needle shapes is of significant interest. In the pharmaceutical industry, opportunities to modify existing processes are often more limited than in the production of other solids, as the polymorph, purity, and potential solvents may be fixed during the process design stages due to regulatory and safety requirements. Furthermore, for new small molecule APIs, research and development costs (including clinical trials) typically exceed 1 billion U.S. dollars per compound, with as few as six years of patent protection remaining after launch.2 Therefore, it is essential that approaches in crystal-shape engineering in this industry are designed to fit within current development schedules. In this article a “turn-key” approach is described for identifying and classifying needles. It is worth noting that although we restrict the description of needles to crystals with aspect ratios of (1:1:≳100), crystals with aspect ratios of (1:1:≳10) may still be “troublesome” during subsequent processing operations and more equant shapes may be desired. Molecules with specific arrangements in the solid state (i.e., polymorphs) that form needles upon crystallization can be categorized as “absolute” or “conditional” needles, based on © 2013 American Chemical Society

whether they form needles in all solvents (absolute) or only in a specific set of solvents (conditional). These equivalence classes were first proposed by Sizemore, with examples of both provided in ref 3. The need to specify both molecule and structure within this categorization is exemplified by cocrystals of carbamazepine and isonicotinamide (CBZ-INA). Although the crystallization of form II of CBZ-INA results in needles, the crystallization of form I does not.4 The classification of a molecule and polymorph as “likely to form” an absolute or conditional needle can be used to guide subsequent research and development efforts. For brevity, the term “likely to form” has been omitted throughout this article, and when referencing a specific system the term “molecule” is used to imply both the atomic constituents and the polymorph. Crystal shape should not be evaluated as a metric in determining an ideal solvent (or combination of solvents) for absolute needles. For absolute needles, efforts to “abate” needles (i.e., to form crystals with more equant shapes) should instead be geared toward mechanical (e.g., milling) or nonchemical (e.g., thermal cycling) routes. Conversely, the ability to abate needles on the basis of solvent selection merits consideration for systems categorized as likely to form conditional needles. This classification requires that the polymorph obtained (including solvates) is independent of the solvent system. Received: December 14, 2012 Revised: June 14, 2013 Published: June 17, 2013 3341

dx.doi.org/10.1021/cg301830u | Cryst. Growth Des. 2013, 13, 3341−3352

Crystal Growth & Design

Article

Figure 1. Schematic demonstrating the condition Rtips ≳ 100 × Rsf required for a steady-state growth shape classified as a needle. The shape is elongated in the b direction, with the {110} and {010} families classified as tips. In (a), a needle is formed with R{010} > R{110}. (Note: the aspect ratio in (a), is ≈1/5 the typical aspect ratio of a needle-shaped crystal). In (b), the significantly decreased relative growth rate of the {010} family of faces results in a more equant steady-state growth shape.

Figure 2. Schematic demonstrating the scenarios for absolute perpendicular growth rates that form needles, which in this case are elongated in the b direction. The green-, yellow-, and red-colored faces indicate “bulk transport limited”, typical, and sparingly slow growth rates, in this article, scenario 2 is evaluated.

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STEADY-STATE GROWTH SHAPES The shape of a growing crystal, with constant perpendicular growth rates for each face, evolves toward a unique and stable steady state (see refs 1 and 6). Steady-state crystal growth shapes can be constructed from the relative perpendicular growth rates of faces by applying the Frank−Chernov condition, given by

These statements are offered as recommendations based on this approach, noting that during growth there can be many additional factors impacting shape (and other key process characteristics such as yield), including solubility and impurity rejection that are often influenced by solvent selection. Solubility directly impacts the time required for steady-state crystal growth shapes to be obtained. Therefore, although the choice of solvent will not affect the steady-state shape of a system classified as an absolute needle, such crystals when grown in one solvent may be more or less needlelike than in another (with different solubility), if growth is stopped after the same amount of time or process yield. Furthermore, altering the amount, shape, and surface area of seed crystals may also influence the progress made toward the steady-state growth shape in a given process. In this article, we develop and support a causality for the formation of needles at low supersaturations that is based on the Burton, Cabrera, and Frank spiral growth mechanism.5 This article provides three case studies that demonstrate the application of this approach for identifying and categorizing PBC networks as likely to form conditional or absolute needles and concludes with general insights into solvent selection for other systems based on the understanding developed from the general methodology and cases presented.

Ri = constant xi

for i = 1, ..., N

(1)

where Ri and xi are the relative growth rate and distance from a common point within the crystal to face i, respectively.7−9 As a consequence of eq 1, the steady-state growth shape of a crystal is determined by the perpendicular growth rates of the slowest growing set of faces that form a bounded convex polyhedron.9 Eq 1 applies to the stable growth of single crystals (i.e., no dendrites or agglomerates). For needle-shaped crystals, each face within this slowest growing set (i.e., those present on the steady-state growth shape) can be classified as belonging to one of two groups. These groups are hereafter denoted as “tips” and “surrounding faces.” Tips are faces with outward normal directions containing a component parallel to the elongated direction (axis) of the crystal; no such component exists for surrounding faces. For example, if a crystal forms a needle that is elongated in the b 3342



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direction, surrounding faces are those denoted using Miller indices as (h0l), while tips are those with (hkl), where k ≠ 0. For brevity, the term “needle direction” is used to indicate the direction of elongation. In order for the steady-state shape of a growing crystal to have an aspect ratio consistent with a needle shape (i.e., aspect ratios given by ≈1:1:100−1000), Rtips ≳ 100 × Rsf, where Rtips and Rsf are the relative growth rates for faces grouped as tips and surrounding faces, respectively. This is demonstrated in Figure 1. Growth Rates Resulting in Needles. Equation 1 allows for three possible scenarios (denoted as scenarios 1, 2, and 3) with regards to the absolute perpendicular growth rates of faces on crystals that form needles. In scenario 1, surrounding faces grow by spirals at “typical” rates, while tips grow by a roughened growth mechanism at diffusion-limited rates (see the Supporting Information of ref 1). In scenario 2, surrounding faces grow by spirals at “sparingly slow” rates, with tips growing at typical rates (surface integration limited) by either a spiral or two-dimensional (2D) nucleation mechanism. In scenario 3, surrounding faces grow by spirals at sparingly slow rates, with tips growing by a roughened growth mechanism at diffusion limited rates. These scenarios are shown in Figure 2. In the literature, there has been a notable bias toward the causality offered in scenario 1, with efforts geared at explaining the fast growth of tips. To that end, Chen and Trout recently stated that “there are no suitable kinetic-based models for the fast-growing needle tips.”10 Furthermore, the assumption is often made that tips can be categorized a priori as S or K faces, using PBC theory (see ref 1), while attempting to provide the causality for the formation of needles in specific cases. Following this assumption would imply that tips grow by a roughened mechanism at any supersaturation. If this assumption were universally true, none of the faces referred to in this article as tips could contain two or more PBCs. It is shown that this was not the case for any of the systems mentioned in this article. Furthermore, other authors have found tips containing two or more PBCs (see ref 3 and Table 2 of ref 11). More detailed attempts to explain the growth of needles in specific cases have often focused on the growth of a single tip face and applied nonmechanistic models (e.g., the attachment energy model) to connect thermodynamic properties measured through simulations to a shape prediction. Although Chen and Trout were able to demonstrate the value of these predictions in terms of generating a solvent mixture a priori to abate needles for a specific case, their estimated aspect ratios were atypical of needle-shaped crystals (1:1:≈1.3−1.8).10 In this article, the causality offered in scenario 2 is evaluated. In this scenario, surrounding faces grow by spirals at sparingly slow growth rates, while tips grow by spirals or 2D nucleation and growth typical rates. However, within this article, the scenario is further limited to cases wherein all F faces present on the crystal’s steady-state growth shape, including tips, grow by spirals. Therefore, the aspect ratios calculated in this article are valid only for system-specific (nonzero) regions of “low” supersaturations, which can be determined by following the analysis presented in ref 12. It can be shown that at higher supersaturations (i.e., supersaturations at which some or all tips grow by 2D nucleation), larger aspect ratios will be determined as steady-state shapes.

Figure 3. AFM image of a growth spiral on a surrounding face of a needle shaped 16.16.16 triacylglycerol crystal.18 The crystal is elongated in the [010] direction. (Note: see ref 18 for scale bars). The shape of this spiral agrees with the shape of a spiral located on a surrounding face that is expected based on the causality for needleshaped crystals presented in this article (i.e., the elongated direction of the crystal is identical to the very slow moving edge of the spiral). (Reprinted with permission from ref 18. Copyright 2002 Elsevier.)

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SPIRAL GROWTH KINETICS In spiral growth, screw dislocations act as renewable sources of edges. These edges are thermally roughened such that they provide a source of kink sites at which molecules can incorporate into the crystal (see refs 1, 5, 13, and 14). The perpendicular growth rate of faces growing by spirals is given by (2) G = hτ −1 where h and τ are the height of an edge (also referred to as the step height) and characteristic rotation time of a spiral on the face. The rotation time, τ, is equivalent to the time required for the innermost turn of this spiral to complete one revolution; it is also the time lapsed between the passage of successive steps at any fixed point on a crystal face (assuming a single dominant spiral). For N-sided polygonal spirals composed of edges growing with “on/off” velocity profiles, N

τ=

∑ τî

(3)

i=1

where τ̂i is the time required for edge i, which is pinned at its emergence on the face by the dislocation and not moving in its outward normal direction, to reach its critical length, lc,i. This summation consists of the edges surviving the first turn of the spiral.15 As edge i does not move in its outward normal direction during this stage (vi = 0 for li < lc,i) and edge i + 1 has yet to appear on the face, edge i lengthens by the outward normal motion of the preceding edge, denoted as edge i − 1. The velocity with which edge i lengthens, which is referred to as its tangential velocity, has been determined by Snyder and Doherty15 as vi − 1 vit = sin(αi − 1, i) (4) during this stage, where αi−1,i is the interior angle between edges i − 1 and i, and vi−1 is the steady-state velocity of edge i − 1 (vi−1 for li−1 ≥ lc,i−1). Therefore, τ̂i is given by τî = 3343

lc, i sin(αi − 1, i) vi − 1

(5)



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Figure 4. Hypothetical needle-shaped crystal elongated in the (a) b direction and a corresponding spiral on a (b) (100) face.

Figure 5. Histograms of γstot, γsd, and γsa for the 132 solvents listed by Barton,25 with curves corresponding to fitted probability densities based on the normal distribution. In (a) and (c), there are twelve and four compounds with larger values of γstot and γsa than 35 erg/cm2, respectively, which are not shown. In (a) and (c), bi- and trimodal distributions were determined, with the populations in (c) corresponding to poor, moderate, and strong hydrogen bonding as determined by Barton.25

Substituting eq 5 for τ̂i into eq 3, the characteristic rotation time of a spiral is given by N

τ=

∑ i=1

where ae,i is the distance between growth units along edge i, ϕkink is the work required to form a kink site along edge i, σ is i the dimensionless supersaturation defined as σ ≡ Δμ/kbT ≈ ln(CS/Ceq) ≈ (CS/Ceq) − 1, where CS and Ceq are the concentrations of solute in solution at the surface and at equilibrium, respectively, Δμ is the chemical potential difference between a solute molecule in solution and in the crystal, kb is the Boltzmann constant, and T is the temperature. The steady-state velocity of edge i is given by

lc, i + 1 sin(αi , i + 1) vi

(6)

where αN,N+1 = αN,1 and lc,N+1 = lc,1. Predicting τ a priori requires the ability to determine lc,i and vi for the N edges surviving the first turn of the spiral.15 Closed-form expressions for critical lengths and edge velocities are derived in refs 16 and 14, respectively, and provided here. Assuming that edge i obeys the Gibbs− Thomson law, its critical length is given by 15

kink ⎛ ae , i ⎞⎛ ϕ ⎞ lc, i = 2⎜ ⎟⎜⎜ i ⎟⎟ ⎝ σ ⎠⎝ k bT ⎠

⎛ k− ⎞ vi ≈ ap , iσ ⎜ ⎟ ⎝ ni̅ ⎠

(8)

where ap,i is the distance traveled by edge i as the result of the incorporation of a new row of molecules, k− is the rate constant for detachment of a molecule from a kink site, n̅i is the average number of edge sites separating successive kinks, and the

(7) 3344



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substitution of σ ≈ (CS/Ceq) − 1 has been made. Applying the single-site model developed by Frenkel (see refs 17 and 14) to determine n̅i, results in kink x0, i 1 ⎛ϕ ⎞ = exp⎜⎜ i ⎟⎟ + 1 ni̅ = ae , i 2 ⎝ k bT ⎠ (9)

associated with this approximation is minimized as the ratio of kink {[exp(−ϕkink D,tip/kbT)]/[exp(−ϕD,sf /kbT)]} → ∞. In light of eq 13, the following causality is offered for the growth of needle-shaped crystals. The growth of a needle requires a single dominant PBC, denoted by a ★, within the network of PBCs that results in

where x0,i is the average distance between successive kink sites adjacent to edge i.17 Spirals on surrounding faces are expected to be composed of two opposing parallel edges, which are also parallel to the needle axis and, hereafter, denoted by the subscript D for which vD ≪ vi≠D, and eq 3 becomes

(14)

τ ≈ 2τD̂ + 1

kink ϕ★kink ≳ ϕi ≠★ + 5k bT

for the molecule in question. In order to provide a means of reference, 5kbT ≈ 2.1 × 10−13 erg at 300 K (or 5RgT ≈ 3 kcal/ mol, where Rg is the universal gas constant). For needles conforming to this causality, PBC★ will be contained in all surrounding and cannot be contained in any tips. Therefore, PBC★ must run parallel to the needle direction. For example, if PBC★ is parallel to the b direction, tips will be described as (hkl), where k ≠ 0, using Miller indices. The existence of PBC★ within all surrounding faces results in relative perpendicular growth rates that are much slower than those expected for faces that are perpendicular to the needle direction. Furthermore, the presence of PBC★ on surrounding faces is expected to led to spirals on these faces that mimic the shape of the crystal (i.e., they are anisotropic and elongated in the same direction as the crystal). An example of a spiral conforming to this expectation is shown in Figure 3. Substituting eq 14 for ϕkink ★ in eq 13 results in

(10)

Equation 10 indicates that on spirals with two comparatively slow moving (dominant) edges, the most time-consuming process in each revolution is the lengthening of the edges immediately following D to their respective critical lengths. A spiral consistent with expectations for such a case that was observed experimentally on a surrounding face of a needle is shown in Figure 3. These expectations are illustrated in the schematic shown in Figure 4. Substituting τ̂D+1 from eq 5 into τ from eq 10 and the resulting expression in eq 2 yields v G≈h D 2lc, D + 1 (11)

R tip R sf

where αD,D+1 ≈ π/2 is assumed. The order of magnitude of the error introduced in predicting G using these simplifications is ≈h∑iN= 1τ̂−1 i≠D. Although often unacceptable in shape predictions for equantly shaped crystals (e.g., the case of naphthalene presented in ref 12), for high aspect ratio shapes this error is likely to be insignificant. Substituting the eqs 7, 8, and 9 into eq 11, results in kink ⎛ kink ⎞−1 hσ 2 ⎛ ap , D ⎞⎜ ϕD + 1 ⎟ − ⎛⎜ ϕD ⎞⎟ ⎜ ⎟ k exp⎜ − G≈ ⎜ ⎟ ⎟ 2 ⎝ ae , D + 1 ⎠⎜⎝ k bT ⎟⎠ ⎝ k bT ⎠

(12)

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WHY DO NEEDLES FORM? Most molecular organic crystals are arranged in lattices with orthorhombic or lower symmetry,19 with molecules “held in place” through an anisotropic network of intermolecular interactions.20 These interactions form the PBCs that “run” throughout the crystal and are manifested in the respective ϕkink i present on the different exposed faces of a growing crystal. Recalling that Rtip ≳ 100 × Rsf is required for a needle to form as a steady-state growth shape, and the (dominant) proportionality G ∝ n̅−1 D , the relationship given by R sf

≈

exp( −ϕDkink /k bT ) ,sf

exp( −ϕ★kink /k bT )

= exp[6(5)] (15)

As a consequence of eq 15, the ability to abate needles for a specific system through solvent selection requires that a solvent kink exists that minimizes ϕkink ★ relative to ϕi≠★ to such an extent that a needle is no longer formed. From this, the classification of a molecule as an absolute needle emerges as an instance where solvents (or combinations thereof) do not exist with this ability. Conversely, a system is a conditional needle if such solvents do exist. Differing from the work of others, the causality for the formation of needles presented in this article does not exclude tips from being classified as F faces (see ref 3), nor does it restrict them to growing by either a two-dimensional or rough growth mechanism.21 In applying this causality for the formation of needles, faces categorized as tips will grow by spirals in so much as σ ≤ σ2D,hkl; where σ2D,tip defines the nonzero supersaturation beyond which the face grows by a 2D nucleation mechanism (see ref 12). With the use of this model, only the work required to form kink sites along the slowest moving edges on each face present on the crystal’s steady-state spiral growth shape, ϕkink D , are required to predict whether a molecule will form a needle. Macroscopic relationships, similar to those developed for γedge i in ref 12, can be used to determine ϕkink along edge i in i solution. For vapor grown crystals, the work required to form a kink on a unit area basis is given by

after simplification. For ≫ kbT, the proportionality given −1 by G ∝ exp(−ϕkink D /kbT) or G ∝ n̅D becomes the dominant feature of eq 12.

exp( −ϕDkink /k T ) ,tip b

exp[−(ϕ★kink − 6(5)k bT )/k bT ]

≈ 150

ϕkink D

R tip

≈

≳ 100

cr γi kink = γtot, =− i

(13)

1 Etot, i 2 si

(16)

γcrtot

where is the “total” surface energy at a kink site (per unit area), Etot is the total (London dispersion + electrostatic + hydrogen bonding, etc.) strength of the intermolecular interactions in the crystal that are broken to form a disturbance,

emerges as a condition required for the growth of needles. In this analysis, the ratios of the remaining terms in eq 12 are approximated as unity in comparison to the ratio of exponential terms for faces with large differences between ϕkink D . The error 3345



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and si is the surface area exposed at a kink site along edge i. The factor of a half that is introduced in eq 16 has been treated elsewhere.5,17 By convention, Etot < 0 and γtot > 0 for attractive (favorable) interactions. The conversion between γkink and ϕkink i i is given by ϕi kink = γi kinksi

where ha and hd are the hydrogen-bond accepting and donating components of the intermolecular interactions, respectively, and γcrz,i = −(1/2)(Ez,i/si). (Note: Wkink,z < 0 for ad,i repulsive solid-side interactions.) In practice, several common force fields, which have been used extensively to characterize interactions in the solid state, do not determine specific and separable polarization and hydrogen-bonding interactions. Instead these interactions are often “lumped” in the Coulombic interactions present due to the partial charges assigned to individual atoms. Therefore, in lieu of determining these interactions and assuming hydrogen bonding is accounted for in the Coulombic interactions within the crystal, these interactions are “mixed” with the associative interactions within the solvent, and Wkink,z ad,i is approximated as

(17)

Therefore, for vapor grown crystals becomes

ϕkink i

= −Etot,i/2 and eq 14

|Etot,★| ≳ |Etot, i ≠★| + 10k bT

(18)

Although eq 18 can be used to predict whether a molecule is likely to form a needle when grown from the vapor, further modifications are necessary for solution-grown crystals. However, without these modifications, eq 18 provides the ability to characterize a molecule as a conditional needle, as there is likely to be a solvent that does not significantly decrease kink ϕkink ★ relative to ϕi≠★. Solvent Effects and Variability. In the predictive growth models developed by Doherty et al.,15,22−24 classic bulk interface approximations are used to determine γkink and ϕkink i i kink for solution-grown crystals. In these models, γi , is given by cr s kink γi kink = γtot, + γtot − Wad, i i

kink Wad, i

Wkink ad,i

where is the surface energy of the solvent, and is the work of adhesion between the crystalline and solution phases. Using eq 19, the work required to form a disturbance along edge i can be approximated as kink s ϕi kink = γi kinksi = −Etot, i /2 + γtot si − Wad, i si

(20)

There are three anisotropic parameters in eq 20; namely, si, Etot,i, and Wkink ad,i . The intermolecular interactions and geometry of kink sites, which determine Etot,i and si, respectively, are fixed for a given molecule. Therefore, Wkink ad,★ becomes the “targeted” anisotropic parameter that must be maximized relative to Wkink ad,i≠★ by a certain solvent (or combination of solvents) for solvent selection to effectively abate the growth of needleshaped crystals for a given system. The work of adhesion across an interface includes contributions from dispersive, polar, and hydrogen-bonding interactions, occurring between the solid and solution.25 Wkink ad,i can be treated as a summation given by kink Wad, i =

∑

1/3 2 γzs = εVM δz

(21)

where d, p, and h denote the London dispersion, polar, and hydrogen bonding components of the intermolecular interactions. The traditional geometric mean approximations for each component of Wkink ad,i result in kink,d Wad, ≈2 i

kink, p Wad, i

≈2

kink, h Wad, ≈2 i

γd,cri |γd,cri|

γpcr, i |γpcr, i|

(|γd,cri|γds)1/2

cr γha, i cr |γha, | i

cr s 1/2 (|γha, |γ ) + 2 i hd

δa ≡

(23) cr γhd, i cr |γhd, | i

cr (|γhd, |γ s )1/2 i ha

|γc,cri |

(|γc,cri |γas)1/2

(25)

(26)

δ h2 + δp2

(27)

The variability in surface energies for the 132 solvents whose solubility parameters are listed by Barton25 is shown in Figure 5 (see also Figure 10). These energies were determined using eq 26 and 27, assuming ε = 0.1. The 132 solvents shown in Figure 5 are used in this article to indicate the range of surface energies for all “common” solvents. In Figure 5, a large degree of variability is apparent between the strength of associative interactions determined for these solvents, when compared to dispersive interactions. Additionally, there are very few solvents with γsd ≳ 25 erg/cm2. It is worth noting that this analysis excludes “exotic” solvents

(22)

(|γpcr, i|γps)1/2

+2

γc,cri

where ε is a factor accounting for the presence of an interface (typically ε ≈ 0.1),26 VM is the molar volume of the solvent, and δz is the solubility parameter associated with the component z, with z = d, p, or h.25 The solubility parameter corresponding to associative interactions can be determined from25

kink, z Wad, i

z = d,p,h

2 cr (|γd,cri|γds)1/2 |γd, i|

where the subscripts a and c denote the associative and Coulombic components of the surface energies. Equation 25 allows for the rapid determination of Wkink ad,i , using pairwise intermolecular interaction parameters from various force fields to calculate the solid-side contributions and tabulated solubility parameter data25 for the solution side. With the use of eq 25 to determine Wkink ad,i , a “trade-off” is made, where more accurate methods for calculating interfacial energies (i.e., one that is obtained using a molecular simulation method)10 are bypassed in favor of an approach that can rapidly be implemented. kink If the minimization of ϕkink ★ relative to the remaining ϕi≠★, kink through the relative maximization of Wad,★ is taken as the approach for abating needles, eq 25 provides a practical guide for the system-specific choice of solvent. For a system with large Ed,★ relative to Ed,i≠★, a solvent that has mostly dispersive interactions should be chosen (e.g., heptane). Conversely, for systems with a large Ec,★ relative to Ec,i≠★, a strong polar/ hydrogen-bonding-capable solvent should be used (e.g., methanol). Components of the surface energies for the solvent side can be approximated using solubility parameters. Barton divides these parameters into their dispersive, polar, and hydrogenbonding components.25 The relationship between surface energy and solubility parameter is given by

(19)

γstot

≈

γd,cri

(24) 3346



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such as ionic liquids, or supercritical CO2, which may have quite different properties. Furthermore, salt/surfactant, pH, and thermal effects on the surface energies of the solvent are not considered. The implications of solvent variability, with regards to the classification of needles as absolute or conditional, are discussed in the following section. Classification of Needles. In this section, an a priori classification of molecules that are likely to form absolute or conditional needles is presented. This classification is based on the recommendations from Solvent Effects and Variability and the variability in γsz for common solvents shown in Figure 5. For both cases, only the set of common solvents listed by Barton25 were considered. An absolute needle is predicted if the PBCs in the crystal are either all electrostatic or all dispersive in nature, and there is an exceptionally large (≳8kbT) difference between PBC★ and the strongest interaction present in tips. This prediction arises from kink the inability to maximize Wkink ad,★ relative to Wad,i≠★ in a given solvent, to an extent that the steady-state growth shape is no longer a needle. (Recall: needles are described as having aspect ratios ≈ 1:1:100−1000). Furthermore, an absolute needle is predicted if a solvent with γsd ≳ 25 erg/cm2 is required to obtain non-needlelike steady-state shapes. The later condition arises due to the scarcity of such solvents (see Figure 5). Conditional needles are predicted to form if solvents exist that can abate needles. There are two types of conditional needles. In “type I” conditional needles, PBC★ is composed of Coulombic interactions. Type I conditional needles will grow into needle shapes from nonpolar/weak hydrogen-bonding solvents, with growth resulting in more equant shapes in polar or strong hydrogen-bonding solvents (e.g., water, primary alcohols). Alizarin27 and 4-nitro-4′methy benzylidene28 crystal, are examples of type I conditional needles. These systems form needles when grown in nonpolar/non-hydrogen-bonding solvents and more equant shapes when grown in alcohols and other polar and hydrogen-bonding solvents. In “type II” conditional needles, PBC★ is composed of dispersive interactions. Type II conditional needles grow as high aspect ratio shapes from strong polar or hydrogen-bonding solvents, while growth from nonpolar/weak-hydrogen bonding solvents (e.g., alkanes, benzene, and toluene) results in more equant shapes. To apply these classifications using the approach developed in Solvent Effects and Variability requires that the hydrogenbonding interactions present in the crystal are “built into” the force fields used as Coulombic interactions. Furthermore, unique features of molecular organic crystals, such as π-bond stacks that are frequently present within PBC networks, are often not accurately accounted for by common force fields.29 Working within these practical limitations, the ability to determine the PBC networks for a given crystal is the only requirement for predicting and classifying needles. The development of a program for rapidly (≲60 s) determining and visualizing these networks is described in ref 30.

Figure 6. Select PBC diagrams for α-PABA. The growth units are shown in (a), with atoms colored by type (left) and Z J pair (right). In (b−d), circles represent centers of mass colored by the growth unit, and lines represent intermolecular interactions colored by strength. Slices of the (002) face, which is a surrounding face, are shown in (b) and (c), with ξ = 0.0 and ξ = 0.25, respectively. A slice for the (011̅) face, which is a tip, is shown in (d). The intermolecular interaction strengths corresponding to each color are provided in Table 1.

Table 1. Intermolecular Interactions for α-PABA Dimers with Colors Corresponding to Those in Figure 6a line color red mustard green blue magenta

[Z0 J0 Z1 J1] [1 [1 [1 [1 [1 [1

2 1 2 2 1 2

1 1 2 1 2 2

2] 1] 1] 1] 1] 2]

[2 [2 [1 [2

2 1 1 1

2 2 2 2

2] 1] 2] 2]

distance

Ed/kbT

3.86

−26.6

Ec/kbT 0.7

Etot/kbT −25.9

9.52 9.28 13.69 13.69

−6.7 −5.7 −3.8 −3.6

−4.1 −1.3 −0.5 −0.4

−10.8 −7.1 −4.2 −4.0

a

Distance indicates the distance between respective centers of mass (in Å) and T = 300 K. The colors applied to the Z J pairs correspond to those in Figure 6. For brevity, only the Z J pairs in Figure 6 are listed.

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CASE STUDIES The monoclinic polymorph of p-aminobenzoic acid, and the orthorhombic polymorphs of lovastatin and D-mannitol are provided as case studies used to demonstrate the effectiveness of this approach for predicting and classifying needles. In the section Predicted Aspect Ratios and Insights into Solvent Selection, the fidelity of the causality presented in this article for needle classification is further tested by comparing

experimentally obtained growth shapes and predicted aspect ratios. In this section, PBC diagrams are used to display the network of intermolecular interactions occurring for each molecule described in the case studies. The following terminology and methods are used to describe the PBC diagrams contained in the remaining sections of this article: each asymmetric unit within a unit cell is assigned a distinct “Z”, each molecule within 3347



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Figure 7. PBC diagram for orthorhombic lovastatin. The packing arrangement is shown in (a), with atoms colored by the type (left) and asymmetric unit (right). In (b) and (c), circles represent centers of mass colored by asymmetric position and lines represent intermolecular interactions colored by strength. In (b), a slice in the direction of the (210) face, which is a surrounding face, is shown. This slice was taken with ξ = 0.5 and ζ = 0.25. In (c), a slice in the direction of the (011) face, which is a tip, is shown. This slice was taken with ξ = 0.25 and ζ = 0.25. The (011) face contains 3 PBCs and is classified as an F face, which is in agreement with the scenario 2 for needle formation. In (c), the molecules with positions “out of the page” are shown with a thicker stroke, while molecules with positions “into the page” are shown with dashed strokes. The intermolecular interactions corresponding to each color are provided in Table 2.

an asymmetric unit is assigned a distinct “J”, V⊥ refers to the unit vector perpendicular to the face in question, and the position and thickness of each slice are indicated by scalar quantities ξ and ζ, respectively, defined such that the distance along V⊥ from the origin to the center of the slice is given by ξ × dhkl, and for practical reasons ζ = 1/2 corresponds to a slice thickness given by dhkl. On this basis, the distance between a center of mass (located at the position rcom) and the plane corresponding to the center of the slice, splane, is given by scom,plane = |rcom·V⊥ − ξdhkl|

magenta, cyan, brown, purple. These practices are offered here in an effort to establish standards in the construction of PBC diagrams. In the remainder of this section, PBC diagrams are used to test the causality for needle formation given in this article for the set of case studies described. Case Study: α-PABA. In ref 3, the α polymorph of paminobenzoic acid (C7H7NO2), hereafter referred to as αPABA, is characterized as an absolute needle elongated in the b direction. This characterization was based on the experimental observation of needle-shaped crystals for crystals grown in thirteen different solvents, which are listed in Table 4.31 αPABA (CSD ref code: AMBNAC01) crystallizes in the P21/n space group with lattice parameters (a, b, c; α, β, γ) given by (18.551, 3.86, 18.642; 90, 93.56, 90).32 In ref 3, α-PABA was determined to grow as a dimer with 2 dimers per asymmetric unit and 2 asymmetric units per unit cell. With the use of this crystallographic information,32 slices of the PBC network for α-PABA were constructed and are shown in Figure 6. The corresponding interactions are listed in Table 1. These interactions were determined using the same force field33 and partial charge values as in ref 3 for the case of a dimer growth unit. With the assumption of a dimer growth unit for α-PABA in the P21/n space group, the following Z J pairs are possible: 1 1, 1 2, 2 1, and 2 2. In Figure 6, these pairs are colored blue, green, red and cyan, respectively. On the basis of the approach developed in this article, αPABA is classified as a type II conditional needle, indicating that solvents with weak polar and hydrogen-bonding kink interactions are likely to maximize Wkink ad,★ relative to Wad,i≠★, resulting in more equant steady-state growth shapes. Furthermore, as is apparent in Figure 6d, the (011̅) tip face is an F face containing three PBCs, indicating that this face

(28)

and the set of molecules and interactions contained within a slice satisfy the inequality

scom,plane ≤ ζdhkl

(29)

Readers are directed to ref 30 for further details on the construction of these diagrams and the calculation of intermolecular interactions. In the construction of these diagrams the following “best practices” have been identified and were implemented throughout the remainder of this article: (1) centers of mass (and not centroid) positions should be used, (2) centers of mass should be distinguishable for different Z J pairs, (3) summed interactions between molecules with different strengths should be indicated by lines with distinct colors, (4) V⊥ should be used as the viewpoint of the diagram, (5) diagrams should include enough centers of mass that the symmetry operations and the PBCs can be identified, and (6) centers of mass that are located “into” or “out of the paper” should be distinguished. The coloring scheme that was used in this article to indicate interactions with different strengths is given from strongest to weakest by: red, mustard, green, blue, 3348



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Table 3. Intermolecular Interactions for D-Mannitol with Colors Corresponding to Those in Figure 8a line color red mustard green blue magenta cyan brown purple a

Table 2. Intermolecular Interactions for Orthorhombic Lovastatin with Colors Corresponding to Those in Figure 7a Red Mustard Green Blue Magenta Cyan a

[Z0 J0 Z1 J1] [1 [3 [1 [1 [1 [1 [1

1 1 1 1 1 1 1

1 3 3 2 2 3 4

1] 1] 1] 1] 1] 1] 1]

[2 [4 [2 [3 [3 [2 [2

1 1 1 1 1 1 1

2 4 4 4 4 4 3

1] 1] 1] 1] 1] 1] 1]

distance

Ed/kbT

Ec/kbT

Etot/kbT

3.48

−22.1

−1.5

−23.6

11.15 11.39 10.09 10.38 11.19

−4.4 −3.6 −7.9 −9.1 −3.0

−7.5 −7.5 −1.2 0.6 −1.1

−11.8 −11.0 −9.1 −8.5 −4.2

1 3 3 2 3 2 4 3 4

1] 1] 1] 1] 1] 1] 1] 1] 1]

[2 [4 [2 [3 [2 [3 [2 [2 [2

1 1 1 1 1 1 1 1 1

2 4 4 4 4 4 3 4 3

1] 1] 1] 1] 1] 1] 1] 1] 1]

Ed/kbT

Ec/kbT

Etot/kbT

1.3

−27.1

−25.9

5.48 7.89 8.43 6.89 8.69 7.11 9.26

−4.5 −0.5 −2.0 −4.1 −2.2 −1.7 −1.4

−13.0 −8.9 −2.1 0.1 −1.2 −1.3 −1.1

−17.4 −9.4 −4.1 −4.0 −3.4 −3.0 −2.6

For brevity, only the Z J pairs and distances in Figure 8 are listed.

90, 90, 90), one molecule per asymmetric unit, and four asymmetric units per cell (CSD ref code: CEKBEZ).34 The possible Z J pairs within the unit cell are 1 1, 2 1, 3 1, and 4 1. In Figures 7 and 8, these pairs are colored blue, green, red and cyan, respectively. PBC diagrams for the (210) and (011) faces are shown in Figure 7. The interactions shown in Figure 7 are listed in Table 2. These interactions were determined using the generalized amber force field35 with RESP fit partial charges,36 which were assigned using the antechamber.37 Prior to determining the charges, the electronic structure of a single (gas phase) molecule was solved using a density functional theory (DFT) calculation in Gaussian03.38 This calculation was performed using the B3LYP hybrid functional and 6-31G(d) basis set. As shown in Table 2, the PBC network for lovastatin contains a single strongest intermolecular interaction parallel to the c direction. On the basis of the causality for needle formation developed in this article, lovastatin is likely to form a needle elongated in this direction, with the Miller indices for surrounding faces given by (hk0). The strongest interaction in the network, which is red in Figure 7, exists in all surrounding faces and none of the faces in the needle direction. This interaction is mostly dispersive in nature. By contrast, the strongest interaction that exists in faces characterized as tips, which is mustard in Figure 7, has a strong electrostatic component. On the basis of Classification of Needles, these interactions lead to the classification of lovastatin as a type II conditional needle, indicating that solvents with weak polar and hydrogen-bonding interactions kink are likely to maximize Wkink ad,★ relative to Wad,i≠★, resulting in more equant steady-state growth shapes. Case Study: D-Mannitol. D-mannitol (C6H14O6) crystallizes in several different polymorphs. In this case study, the “β” polymorph of D-mannitol was considered. This polymorph is arranged in the orthorhombic P212121 space group with unit cell parameters given by (8.694, 16.902, 5.549; 90, 90, 90), one molecule per asymmetric unit and four asymmetric units per cell (CSD refcode: DMANTL07).39 The possible Z J pairs within the unit cell are the same as for lovastatin. PBC diagrams for the (120), (200), and (011) faces are shown in Figure 8. The interactions shown in Figure 8 are listed in Table 3. These interactions were determined using the same methods applied for lovastatin. As shown in Table 3, the PBC network for D-mannitol contains a single strongest intermolecular interaction, parallel to the c direction. On the basis of the causality for needle formation developed in this article, D-mannitol is likely to form a needle elongated in this direction, with the Miller indices for

Figure 8. PBC diagram for D-mannitol. The packing arrangement is shown in (a), with atoms colored by type (left) and asymmetric unit (right). In (b), the PBC network was sliced in the direction of the (120) face, which is a surrounding face. This slice was taken with ξ = 0.25 and ζ = 0.25. In (c), the PBC network was sliced in the direction of the (200) face, which is a surrounding face. This slice was taken with ξ = 0.0 and ζ = 0.25. In (d), the PBC network was sliced in the direction of the (011) face, which is a tip. This slice was taken with ξ = 0.75 and ζ = 0.25. The intermolecular interaction strengths corresponding to each color are listed in Table 3.

Line color

1 1 1 1 1 1 1 1 1

5.55

distance

[Z0 J0 Z1 J1] [1 [3 [1 [1 [1 [1 [1 [1 [1

For brevity, only the Z J pairs in Figure 7 are listed.

grows by a spiral mechanism at low supersaturations, thereby supporting the causality developed in Why do Needles Form? for needle formation. Case Study: Lovastatin. Lovastatin (C24H36O5) is a LDL cholesterol-lowering drug that was the first in the class of statins to be marketed. This molecule was chosen as a case study due to the large market for statins, and its known experimentally obtained shapes in several different solvents. Lovastatin crystallizes in the orthorhombic P212121 space group with unit cell parameters given by (22.154, 17.321, 5.968; 3349



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Case Studies for each of the solvents in the set of 132 listed by Barton.25 These aspect ratios were approximated by first determining the surface areas exposed at two distinct kink sites on the crystal. These areas were then used in eq 20 to kink kink determine ϕkink ★ and ϕD,tip, where ϕD,tip is the work required to create a kink on the dominant (slowest) edge found on the tip with the slowest perpendicular growth rate. For all cases, the interaction colored mustard corresponds to ϕkink D,tip. The aspect ratios in the elongated direction of the steady-state growth shape, AR, were approximated as

surrounding faces given by (hk0). The strongest interaction, which exists exclusively in surrounding faces, is mostly electrostatic in nature. The strongest interaction that exists in faces characterized as tips, which is mustard in Figure 7, is also mostly electrostatic. On the basis of Classification of Needles, the β polymorph of D-mannitol is expected to form a type I conditional needle with the aspect ratios of the steady-state growth shapes decreasing with increased hydrogen bonding and polar surface energy contributions in the solvent.

■

PREDICTED ASPECT RATIOS AND INSIGHTS INTO SOLVENT SELECTION In an attempt to provide insight into solvent selection, steadystate aspect ratios were predicted for the systems described in

AR ≈

solvent

ϕkink ★

ϕkink D,tip

α-PABA

Δϕ

AR

n-hexane toluene ethyl acetate acetone hexanol DMF acetic acid DMSO 2-propanol acetonitrile ethanol methanol water toluene n-hexane ethyl acetate acetone 2-propanol ethanol methanol water water methanol 2-propanol ethyl acetate n-hexane toluene

5.6 5.0 6.9 7.9 8.0 8.5 9.1 8.6 9.0 9.6 10.1 11.5 17.0 3.9 5.0 4.3 4.6 4.7 5.2 6.0 9.1 4.6 7.5 9.2 11.7 14.8 14.3

2.7 2.0 1.3 1.1 0.7 0.5 1.1 0.4 0.7 0.9 0.7 0.9 1.8 2.9 3.7 1.3 0.8 0.2 0.1 0.1 1.4 1.2 2.4 3.1 4.6 6.9 6.0

2.9 3.0 5.6 6.8 7.3 7.9 8.0 8.2 8.3 8.8 9.4 10.6 15.2 1.0 1.3 3.0 3.8 4.4 5.1 5.8 7.7 3.4 5.2 6.2 7.1 7.9 8.3

15 16 220 740 1200 2300 2400 2800 3300 5200 9800 32000 3.3 × 106 3 5 23 52 100 190 410 2700 37 210 590 1500 3400 4800

lovastatin

D-mannitol

a

Gsf

≈

⎛ Δϕ ⎞ exp⎜ ⎟ hsf ⎝ k bT ⎠

htip

(30)

kink where Δϕ ≡ ϕkink ★ − ϕD,tip. With the use of the approach for rapidly generating PBC diagrams for arbitrary faces, which was described in ref 30, the bounding faces in the needle direction could be determined in each case for each solvent. Predicted aspect ratios for selected solvents in each case are listed in Table 4. Surface energies for a subset of these solvents are listed in Table 5. For α-PABA, this set of solvents included the thirteen experimentally shown to form needles.31 For the case of lovastatin, needle shapes were experimentally demonstrated for crystals grown in 2-propanol and methanol,24 while ethyl acetate and acetone resulted in “rodlike” shapes. The experimental shapes of lovastatin grown in 2-propanol, methanol, ethyl acetate, and acetone are provided in the Supporting Information. Furthermore, D-mannitol has been grown in water with a rodlike shape with AR ≈ 20.40 In Table 4, all of the systems chosen as case studies were predicted to form conditional needles. However, it is worth noting that solubility, or the practicality of certain solvents, was not considered in this analysis. The class denoted as a practical absolute needle is offered to indicate a molecule that forms needle-shaped crystals in all solvents (or cosolvent systems) with reasonable solubility and lacking toxicity concerns (for pharmaceutical compounds). For the three systems considered, in each case, the prediction of needlelike (i.e., AR ≳ 100) or non-needlelike was in agreement with known experimental shapes for the majority of solvents considered. Predicted aspect ratios using this approach disagreed with experimental results for the cases of α-PABA grown in n-hexane and toluene. To explore the range of predicted aspect ratios for each of the case studies, histograms of Δϕ/kbT were constructed for each system constructed over the range of solvents listed by Barton.25 These histograms are shown in Figure 9. The range of values for Δϕ for each system are well-fit by normal

Table 4. Predicted Aspect Ratios for the Systems Described in Case Studies in Selected Solventsa crystal

Gtip

All energies are normalized by kbT (T = 300 K).

Table 5. Values of γsz for Selected Solvents and Corresponding AR for the Case Studies Considered.a AR solvent n-hexane toluene ethyl acetate 2-propanol methanol water

γsd

2

(erg/cm ) 13.4 18.2 13.6 12.5 9.3 7.5

γsa

2

(erg/cm )

γstot

2

(erg/cm )

0.0 0.4 4.4 15.4 26.4 63.5

13.4 18.6 18.1 27.9 35.8 71.1

α-PABA

lovasatin

D-mannitol

15 16 220 3200 32000 3.3 × 106

5 3 23 100 410 2700

3400 4800 1500 590 210 37

a This table demonstrates the “matching” between the relative strength of the dispersive and coulombic components for PBC★ and the relative strengths of these interactions in solvents that prevent needles. Furthermore, it provides a “short list” of solvents that can be used as a basis for experimental design.

3350



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Figure 9. Histograms of Δϕ/kbT for (a) α-PABA, (b) lovastatin, and (c) D-mannitol for the set of solvents listed by Barton.25 Roughly speaking, the aspect ratio of crystal will be ≈1:1:exp(Δϕ/kbT). The curves correspond to fitted probability densities based on the normal distribution with the means and standard deviations listed with units of kbT for each case. Red and green bars correspond to solvents that are likely and unlikely to form needles for each case, respectively.

Figure 10. Scatter plots of γsz for the common solvents listed by Barton.25 Where for the cases of (a) α-PABA, (b) lovastatin, and (c) D-mannitol, the circles are colored such that red and green circles correspond to solvents where growth is likely and unlikely to result in the formation of needles, respectively.

Scatter plots of γsz for the solvents listed by Barton,25 with points acquiring the same colors as the lines in Figure 9, are shown for each case study in Figure 10. These plots demonstrate a correlation between the dominant type of solid-side interaction contained in PBC★, the solvent-side γsd and γsa, and the predicted growth of needle-shaped crystals for a given solute−solvent system. Figure 10 (panels a and b) demonstrate type II conditional needles, where growth in solvents with large γ as results in needles. Figure 10c demonstrates a type I conditional needle, where growth in a solvent with large γsa does not result in needles. For D-mannitol,

distributions. Solvents contained on the left side of these histograms correspond to “favorable” solvents that are likely to abate needles. By contrast, solvents on the right side of this histogram will be “unfavorable” and will likely result in the formation of needles. Of the case studies provided, lovastatin has the most favorable behavior over the range of solvents listed by Barton,25 with a mean value of Δϕ/kbT < 5. Given the mean value of Δϕ/kbT for this system, it is expected that a trial and error approach (ignoring solubility and toxicity concerns) for solvent selection will succeed in providing a solvent that does not result in the growth of needles. 3351



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water was predicted as the only solvent with γsa large enough to abate the growth of needle-shaped crystals.

(9) Zhang, Y.; Sizemore, J. P.; Doherty, M. F. AIChE J. 2006, 52, 1906−1915. (10) Chen, J.; Trout, B. L. Cryst. Growth Des. 2010, 10, 4379−4388. (11) Rinaudo, C.; Boistelle, R. J. Cryst. Growth 1982, 57, 432−442. (12) Lovette, M. A.; Doherty, M. F. Cryst. Growth Des. 2012, 12, 656−669. (13) Markov, I. V. Crystal Growth for Beginners, Fundamentals of Nucleation, Crystal Growth and Epitaxy; World Scientific: Singapore, 2003. (14) Lovette, M. A.; Doherty, M. F. Phys. Rev. E 2012, 85, 021604. (15) Snyder, R. C.; Doherty, M. F. Proc. R. Soc. London, Ser. A 2009, 465, 1145−1171. (16) Lovette, M. A.; Doherty, M. F. J. Cryst. Growth 2011, 327, 117− 126. (17) Frenkel, J. J. Phys. U.S.S.R. 1945, 9, 392. (18) Hollander, F. F. A.; Kaminski, D.; Duret, D.; van Enckevort, W. J. P.; Meekes, H.; Bennema, P. Food Res. Int. 2002, 35, 909−918. (19) Mighell, A. D.; Rodgers, J. R. Acta Crystallogr. A 1980, 36, 321− 326. (20) Dunitz, J. D.; Gavezzotti, A. Chem. Soc. Rev. 2009, 38, 2622− 2633. (21) Meekes, H.; Boerrigter, S.; Hollander, F.; Bennema, P. Chem. Eng. Technol. 2003, 26, 256−261. (22) Winn, D.; Doherty, M. F. AIChE J. 1998, 44, 2501−2514. (23) Winn, D.; Doherty, M. F. Chem. Eng. Sci. 2002, 57, 1805−1813. (24) Kuvadia, Z. B.; Doherty, M. F. Cryst. Growth Des. 2011, 11, 2780−2802. (25) Barton, A. F. M. Chem. Rev. 1975, 75, 731−753. (26) Kaelble, D. H. Physical Chemistry of Adhesion; WileyInterscience: New York, 1971. (27) Algra, R.; Graswinckel, W.; Enckevort, W.; Vlieg, E. J. Cryst. Growth 2005, 285, 168−177. (28) Srinivasan, K.; Sankaranarayanan, K.; Thangavelu, S.; Ramasamy, P. J. Cryst. Growth 2000, 212, 246−254. (29) Paton, R. S.; Goodman, J. M. J. Chem. Inf. Model. 2009, 49, 944−955. (30) Lovette, M. A. Prediction and Modification of Crystal Shapes. Ph.D. Thesis, University of California, Santa Barbara, 2011. (31) Gracin, S.; Rasmuson, A. C. Cryst. Growth Des. 2004, 4, 1013− 1023. (32) Lai, T. F.; Marsh, R. E. Acta Crystallogr. 1967, 22, 885−893. (33) Nemethy, G.; Pottle, M. S.; Scheraga, H. A. J. Phys. Chem. 1983, 87, 1883−1887. (34) Sato, S.; Hata, T.; Tsujita, Y.; Terahara, A.; Tamura, C. Acta Crystallogr., Sect. C 1984, 40, 195−198. (35) Wang, J.; Wolf, R. M.; Caldwell, J. W.; Kollman, P. A.; Case, D. A. J. Comput. Chem. 2004, 25, 1157−1174. (36) Bayly, C. I.; Cieplak, P.; Cornell, W.; Kollman, P. A. J. Phys. Chem. 1993, 97, 10269−10280. (37) Wang, J.; Wang, W.; Kollman, P. A.; Case, D. A. J. Mol. Graphics Modell. 2006, 25, 247−260. (38) Frisch, M. J. , et al. Gaussian 03, revision D.02, Gaussian, Inc.: Wallingford, CT, 2004. (39) Kaminsky, W.; Glazer, A. M. Z. Kristallogr. 1997, 212, 283−296. (40) Ho, R.; Wilson, D. A.; Heng, J. Y. Y. Cryst. Growth Des. 2009, 9, 4907−4911.

■

SUMMARY AND CONCLUSIONS In this article, a rapidly implementable approach for predicting and categorizing molecules as likely to form absolute or conditional needles is presented. This approach is based on the causality for the formation of needle requiring that surrounding faces grow at sparingly slow rates. This work can be confirmed experimentally by monitoring the growth rates of tips and surrounding faces at a variety of length scales, using in situ optical and atomic force microscopy. The approach described in this article can be used to predict steady-state aspect ratios over a broad range of solvents, which can be used to limit the experimental range of solvents tested in designing a crystallization process to specifically avoid the growth of needles. On this basis, a further categorization of conditional needles as types I or II and the general properties of solvents that will result in lower aspect ratio shapes has been provided. For the case studies considered, this approach has demonstrated a remarkable agreement with experimental shapes. Furthermore, for each case study, tips were present that met the “two or more” PBC requirement for classification as F faces. This technique is readily applicable to organic free acid/base chemistries. The extension of this approach to salts and inorganic molecules represents a promising avenue for further development.

■

ASSOCIATED CONTENT

S Supporting Information *

This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge helpful discussions with Jacob P. Sizemore at Cubist Pharmacueticals, Inc. The authors are grateful for financial support provided by Eli Lilly and Companies and the National Science Foundation through CBET-1159746.

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REFERENCES

(1) Lovette, M. A.; Browning, A. R.; Griffin, D. W.; Sizemore, J. P.; Snyder, R. C.; Doherty, M. F. Ind. Eng. Chem. Res. 2008, 47, 9812− 9833. (2) Variankaval, N.; Cote, A. S.; Doherty, M. F. AIChE J. 2008, 54, 1682−1688. (3) Sizemore, J. P. Morphology Change of Molecular Organic Crystals via Molecular Imposters. Ph.D. Thesis, University of California at Santa Barbara, 2008. (4) ter Horst, J. H.; Cains, P. W. Cryst. Growth Des. 2008, 8, 2537− 2542. (5) Burton, W. K.; Cabrera, N.; Frank, F. C. Philos. Trans. R. Soc., A 1951, 243, 299−358. (6) Lovette, M. A.; Muratore, M.; Doherty, M. F. AIChE J. 2012, 58, 1465−1474. (7) Frank, F. C. In Growth and Perfection of Crystals; Doremus, R. H., Roberts, B. W., Turnbull, D., Eds.; Wiley: New York, 1958. (8) Chernov, A. A. Crystallogr. Rep. 1963, 7, 728−730. 3352


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