Crystal Shape Engineering

Crystal Shape Engineering

Michael A. Lovette

Department of Chemical Engineering, University of California at Santa Barbara

Motivation• Crystal shape effects

– Downstream processing efficiency (filtering, flowability, tabletting)• Captured by “broad” shape descriptions (aspect ratio, sphericity)

– End use functionality (catalytic activity, bio-availability)• Depend on relative areas of specific faces

• Crystal shape engineering– Optimizing crystal shapes for the given application by tailoring processing conditions

• Part of the physical property control strategy for APIs

• Factors impacting physical properties of APIs– Internal structure/intermolecular interactions, hydrodynamic environment (slurry

milling), solvent(s), supersaturation, impurities, batch/resonance time

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Academic understanding

Practical understanding

batchsteady-statecoverage timestep advancementattachment/detachment

Figure 1. Typical times for crystallization processes in seconds.

Crystallization Time Scales

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Graduate Research Focus• Fundamental

– Crystal Shape Engineering, Ind. Eng. Chem. Res., 2008, 47, 9812-9833.– Reinterpreting Edge Energies Calculated from Crystal Growth Experiments, J. Cryst.

Growth, 2011, 327, 117-126.– Predictive Modeling of Supersaturation-Dependent Crystal Shapes, Cryst. Growth Des.,

2012, 12, 656-669.– Multi-Site Models to Accurately Determine the Distribution of Kink Sites Adjacent to

Low Energy Edges, Phys. Rev. E, 2012, 85, 021604.

• Application Driven (Today’s Topics)– Crystal Shape Modification through Cycles of Dissolution and Growth: Attainable

Regions and Experimental Validation, AIChE J., 2011, 58, 1465-1474.– Needle-Shaped Crystals: Causality and Solvent Selection Guidance Based on Periodic

Bond Chains, Cryst. Growth Des., 2013, 13, 3341-3352.

… Problematic (Needles) Crystal Shapes

Today’s Talks

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Crystal Shape Modification through Cycles …

Crystal Shape Modification through Cycles of Dissolution and Growth: Attainable Regions and

Experimental Validation

Motivation• Traditional approaches for crystal shape engineering

– Chemical (solvent selection, form/salt/solvate selection, impurities)– Mechanical (slurry milling, dry milling, sonication)

• Non-chemical routes for crystal shape engineering [1,2]

– Snyder et al., developed a model for capturing the effects of performing cycles of consecutive stages of dissolution and growth (cycling) on crystal shapes

– At the time cycling was widely practiced industrially though not adequately modeled

• Understanding system dynamics from fixed point and phase-plane analyses

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[1] Yang, et al., Cryst. Growth Des. 2006, 6, 2799. [2] Snyder, et al., AIChE J, 2007, 53, 1510.

Assuming convex polyhedrons (faceted growth/dissolution) [1]

and defining:

yields

Constant growth/dissolution rates (requires size independence) → independent linear ODEs with

and steady-states at

with

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Shape Evolution: Single Stages

[1] Snyder, et al., AIChE J, 2007, 53, 1510.


and defining:

yields

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Figure 2. Dimensional variables for faceted crystal growth/dissolution.


Figure 3. Dimensionless varibles for faceted crystal growth/dissolution.


and defining:

yields

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• Steady-states are stable and unstable stellar nodes– Equivalent eigenvalues – Orthonormal eigenvectors

• Linear trajectories – N-1 dimensions (N = # faces)– Growth: vector connecting

initial state to steady-state– Dissolution: vector beginning

at the initial state and flowing in the direction opposite to the steady-state

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Shape Trajectories: Single Stages

Figure 4. Representative 2-dimensional trajectories for growth and dissolution.

Recommended Reading: S. Strogatz, Nonlinear Dynamics and Chaos, 1994.

Snyder et al., derive recursive relationship ⇒ independent linear

difference equations:

where,

For b 1

with a single steady-state at

or similarly

where:

Steady-state is a stellar node with

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Cycling Dynamics

lc < 1 Dxg > Dxd stable slow

lc > 1 Dxg < Dxd unstable fast

Properties of steady-state1. Single degree of freedom (line)2. y > 0 ⇒ Dxg > Dxd

3. y < -1 ⇒ Dxg < Dxd

4. xc not between xg and xd

Snyder et al., derive recursive relationship ⇒ independent linear

difference equations:

where,

For b 1

with a single steady-state at

or similarly

where:

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Cycling Dynamics

A

B

P/i0

Same exact behavior for cycling dynamics and loan dynamics

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Fun Fact: System Dynamics in Real Life

Loan dynamics

i < 0 -> bank pays you to borrow (impossible)unstable node (l>1)

Case 1: Out of debt

A

B

P/i0

Case 2: Deeper in debt

Phase Plane Analysis

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Figure Face Dxg Dxd

(a)1 2.9

0.15 0.102 1.2

(b)1 0.1

0.10 0.202 0.3

(c)1 4.4

0.10 0.202 0.3

Figure 5. Representative phase planes for growth, dissolution, and cycling. The entire phase plane can be constructed from the steady-state growth, dissolution and initial shapes.

Table 1. Conditions for Figure 5.

Phase Plane Analysis: Needles

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Figure Face Dxg Dxd

(a)1 -19.1

0.10 0.252 1.0

(b)1 139.7

0.10 0.252 1.0

Figure 6. Representative phase planes for needle shaped crystals. For cycling to be an effective for abating needles requires Rd > Rg and Rd

> x0 in the needle direction.

Table 2. Conditions for Figure 6.

Experimental Setup• A small quiescent crystallizer was designed with a peltier cell used to

rapidly implement stepwise temperature trajectories– Single crystals were observed using an inverted optical microscope

• Saturated solution: (1) T⇓ nucleate, (2) T↑ dissolve, (3) T↓ grow, (4) T↕ cycle– Time-lapse micrographs were used to measure growth/dissolution rates

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Figure 7. Control and setup schematic (a), photograph of peltier cell (b).

Results: Paracetamol• 12 cycles of dissolution and growth: paracetamol in water. Tsat = 22°C.

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Figure 8. Paracetamol in water before (a) and after (b) 12 cycles of dissolution and growth. Scale bars are 200 mm, sg = 0.11, sd = -0.08, Dtg = 75 min, and Dtd = 25 min. A1 and A2 move farther

from the central point (smaller area) , A3 moves closer (larger area).

Axis Gg (nm/s) Gd (nm/s) Rg Rd

A1 8.0 ± 1.3 -19.2 ± 2.5 1.00 ± 0.16 1.00 ± 0.13

A2 6.4 ± 1.3 -15.6 ± 2.2 0.80 ± 0.10 0.82 ± 0.12

A3 6.6 ± 1.4 -21.9 ± 3.3 0.86 ± 0.26 1.17 ± 0.26

Table 3. Absolute and relative growth and dissolution rates, for paracetamol in water. A1 is used as the reference face.

Phase Plane: Paracetamol

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Figure 9. Absolute (a) and relative (b) lengths for paracetamol in water. In (b): squares are increments along the measured trajectory, and circles are increments along the predicted trajectory. The measured values of: a2 = -0.014, a3 = 0.022 and b = 0.97 were used for the prediction. As cycling continues the reference face will become an edge or vertex.

A3

A2

A1

Conclusions• Cycling dynamics were investigated and determined to be

closely related to growth and dissolution dynamics• Cycling may be an effective means to avoid high aspect ratio

crystals only for a limited set of conditions• Experiments using paracetamol in water provided quantitative

agreement with the predicted shape trajectory

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Problematic Crystal Shapes:Causality and Solvent Selection Guidance

Based On Periodic Bond Chains

Problematic Crystal Shapes• Qualitative description of problematic shapes

– Needle shaped crystals “needles” • Typical aspect ratios (a : b : c) = (1 : 1 : ≥100)

– Flat-plate shaped crystals “plates”• Typical aspect ratios (a : b : c) = (1 : ≥100 : ≥100)

• Needles and plates can be problematic during filtration, drying, compaction and other downstream unit operations.

• These shapes are a result of the anisotropic intermolecular interactions found in molecular organic crystals.

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Crystal Shapes• Stable crystals form closed polyhedrons with shapes determined by the relative

growth rates of the slowest growing set of faces– Steady-state growth shape (Frank-Chernov Condition)

• The formation of needles requires all faces with components in the needle direction (tips) to grow ≳ 100 × faster than remaining faces (surrounding faces)

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Figure 2. Rtips 100 × R≳ sf .

b

a c

Figure 3. Two scenarios for the scaling of absolute growth rates for needles.

Face Classification

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• Periodic Bond Chain theory– A continuous and repeatable series of

intermolecular interactions within a crystal form a periodic bond chain

– PBCs will be anisotropic for organic crystals

– PBC networks are polymorph specific

• Faces classified as F, S, or K– Flat - layered, slow - ≥ 2 PBC – Stepped - rough, fast - 1

PBC– Kinked - rough, fast - 0 PBC– The growth of S and K faces is bulk

transport limited

Figure 5. Illustrative PBC diagram.

Figure 6. F, S, and K faces for a crystal with 3 periodic bond chains.

Lovette, et al., IECR, 2008.

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Identifying PBCs• One approach is visual identification of PBCs in molecular

organic crystals– Input: atomic positions, partial charges, crystallography

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Color Etot (kbT)-1

Red -6.8

Cyan -5.6

Yellow -2.4

Green -0.8

Figure 7. Periodic Bond Chain diagrams for naphthalene (P21/a). Interaction strengths were calculated using the generalized amber force field with RESP fit charges. T = 300K

(00-1) (11-1)

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Spiral Growth

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• Spiral sides are edges (steps) formed parallel to PBCs

• Steps advance with velocity v, to complete a layer in time ts

Figure 8. Layered growth variables.

Figure 9. Top view of the first revolution on a square spiral. ti = 1-3.

[1] Snyder and Doherty, Proc. Roy. Soc. A, 2009.

Rotation Time [1]

Spiral Growth

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• Spiral sides are edges (steps) formed parallel to PBCs

• Steps advance with velocity v, to complete a layer in time ts

Figure 8. Layered growth variables.

Calculating Kink Energies• Bulk interface model used to approximate kink energies

• gic and Wi

ad are site dependent (determine shape)– Assuming hydrogen bonding in the crystal is “lumped” into the

electrostatic component

• gic,j = ½ Ej,i

• Solubility parameters provide estimates for g s,j [1]

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[1] Barton, Chem. Rev., 1975, 75, 731.Ej,i is the j component of the potential energy between neighbors in direction i for the solid.

Needles• Origin

– A single PBC with fkink ≈ 5 kbT greater than remaining PBCs

• Growth rate scaling– All surrounding faces contain PBC and grow at a sparingly slow rate– Tips do not contain PBC and grow at a typical rate

• A priori categorization– Absolute: Most likely for the same “variety” of interactions in PBC«

and remaining PBCs and large differences between Ez, and Ez,i≠

– Conditional: Two types of conditional needles• Type 1: PBC« contains mostly electrostatic interactions

– needles in non-polar/non-hydrogen bonding solvents (e.g., heptane)• Type 2: PBC« contains mostly dispersion interactions

– needles in strong polar/hydrogen bonding solvents (e.g, alcohols)

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Spiral Growth Model for Needles

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• PBC results in “dominant” slow moving parallel edges on surrounding faces

• Dominant spiral side in agreement with the needle direction

Figure 10. Growth spiral on needleelongated in the [010] direction. [1]

[1] Hollander et al., Food Res Int, 2002, 909.

Rotation Time (needles)

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PBC diagrams: lovastatin

Line Color Ed (kbT)-1 Ec (kbT)-1 Etot (kbT)-1

Red [001] -22.1 -1.5 -23.6Mustard -4.4 -7.5 -11.8Green -3.6 -7.5 -11.0Blue -7.9 -1.2 -9.1Magenta -9.1 0.6 -8.5

(c)(a) (b)

b

c

a

Figure 11. Unit cell (a), (210) face (b), and (011) face (c) for the orthorhombic

P212121 structure of lovastatin. Lovastatin forms a needle elongated in

the c direction.

Figure 12. Scatter plot of g a,g d for the 132 solvents listed by Barton.[1] Green and red points correspond to Df < 5 and

Df > 5, respectively.

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Results: lovastatinSolvent f fD,tip Df ARtoluene 3.9 2.9 1.0 3hexane 5.0 3.7 1.3 5EA 4.3 1.3 3.0 23acetone 4.6 0.8 3.8 52IPA 4.7 0.2 4.4 100ethanol 5.2 0.1 5.1 190methanol 6.0 0.1 5.8 410water 9.1 1.4 7.7 2700

Table 1. Predicted kink energies normalized by kbT and aspect ratios for

selected solvents. The highlighted values correspond to known cases.

[1] Barton, Chem. Rev., 1975, 75, 731.

Experimental Shapes: lovastatin

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Figure 14. Experimental shapes of lovastatin grown in (top right

to bottom left) ethyl acetate, acetone, methanol, IPA.

(Courtesy Z. Kuvadia, UCSB)

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Results: a-PABA

Table 2. Predicted kink energies normalized by kbT and aspect ratios for the 13 solvents shown to result in the

formation of needles. [1]

Solvent f fD,tip Df ARhexane 5.6 2.7 2.9 15

toluene 5.0 2.0 3.0 16

EA 6.9 1.3 5.6 220

acetone 7.9 1.1 6.8 740

hexanol 8.0 0.7 7.3 1200

DMF 8.5 0.5 7.9 2300

acetic acid 9.1 1.1 8.0 2400

DMSO 8.6 0.4 8.2 2800

IPA 9.0 0.7 8.3 3300

acetonitrile 9.6 0.9 8.8 5200

ethanol 10.1 0.7 9.4 9800

methanol 11.5 0.9 10.6 32,000

water 17.0 1.8 15.2 3.3E6

Figure 15. Scatter plot of g a,g d for the 132 solvents listed by Barton. Green and red points correspond to Df < 5 and Df

> 5, respectively.

[1] Gracin et al., Cryst Growth Des, 2004, 1013.

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Results: d-mannitol

Solvent f fD,tip Df ARwater 4.6 1.2 3.4 37methanol 7.5 2.4 5.2 210IPA 9.2 3.1 6.2 590EA 11.7 4.6 7.1 1500hexane 14.8 6.9 7.9 3400toluene 14.3 6.0 8.3 4800

Figure 16. Scatter plot of g a,g d for the 132 solvents listed by Barton.[1] Green and red points correspond to Df < 5 and

Df > 5, respectively.

[1] Barton, Chem. Rev., 1975, 75, 731. [2] Ho, et al., Cryst Growth Des, 2009, 4907.

Table 3. Predicted kink energies and aspect ratios for selected solvents.

Flat Plates – Work in Progress• Origin

– Two symmetry related PBCs (PBC1 and PBC2) with fkink ≈ 5 kbT greater than remaining PBCs.

• Growth rate scaling– Slow growing faces contain both PBC1 and PBC2 and grow at a

sparingly slow rate– The remaining faces may contain either PBC1 or PBC2 but not both

– these faces will grow at a typical rate.• A priori categorization

– The same categorization exists as for needles. – The same process is proposed for investigating solvents.

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Hypothetical Plate System• PBC1 and PBC2 exist together only in {001} faces• PBC1 and PBC2 exist individually in other faces

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Color Etot (kbT)-1

Red [110, -110] -17.0

Cyan -5.6

Yellow -2.4

Green -0.8

Figure 17. Periodic Bond Chain diagrams for hypothetical plate forming system (centrosymmetric, P21/a). Due to the symmetry of this system, the Red interactions

will only be found together on {001} faces.

{001} (11-1)

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Hypothetical Plate: Shape Predictions

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Color Etot (kbT)-1

Red [110, -110] x

Cyan -5.6

Yellow -2.4

Green -0.8

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x = -6.5 x = -8.5 x = -10.5 x = -17.5

Figure 18. Predicted steady-state vapor growth shapes for hypothetical system - as the strength of the red interaction increases the plate thickness decreases.

Conclusions and Future Work• Criterion for rapidly identifying and categorizing the potential

for a specific molecule/polymorph to form needle or flat-plate shaped crystals have been established.

• a-PABA, lovastatin, and d-mannitol were categorized as type II, type II, and type I conditional needles, respectively.

• Anisotropic shapes which are problematic to process form due to anisotropic intermolecular interactions within molecular organic crystals. Guided solvent selection may be able to result in more isotropic steady-state shapes.

• Case studies and further work is needed for flat plates.

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Crystal Shape Engineering