1

Pathways to Crystal Nucleation and Growth

Andrea Robben BrowningGlenn H. FredricksonMichael F. Doherty

Complex Fluids Design ConsortiumAnnual Meeting

2

Background

• Organic crystals are found in everyday products and structure is critical to their usefulness

• Many molecules can form two or more solid structures (polymorphs)

• Polymorphs can have different properties (solubility, strength, etc.)

• Nucleation is the initial stage of polymorph formation

3

Motivation

• Classical Nucleation Theory traditionally used– Nucleus shape and structure are inputs – Critical size is the output

• Appropriate surface energy at nucleus size scales is unknown

• Surface stress may also play a role• Gain understanding of nucleation mechanism for

development of nucleation theory• Current work

– Colloids– Stearic Acid in Hexane

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Colloid Particles

• Hard-core screened repulsive Yukawa pair potential • Solvent treated implicitly• Molecular Dynamics simulations

– NVT ensemble

– 32,000 particles in box (tested with 62,000)

• Seeded initial conditions– FCC or BCC polymorph seed– Variable seed size

• Unseeded, homogeneous, fully random initial conditions (for comparison with Weitz data)

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Yukawa Potential

σ = hard core diameter, 1/κ = screening length, ε = interaction energy when particles touch

σσκσε

r/

)]1r/(exp[u/

−−=

0

0.5

1

0 0.5 1 1.5 2 2.5 3

Screening length decreasing, 1/κσ dec.

pair

pote

ntia

l (u/

ε)

pair separation (r/σ)

Hard core

Large screening length

Small screening length

∞=εu/

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Phase Diagram for Weitz ExperimentsA. Hynninen and M. Dijkstra, Phy. Rev. E, 68, 021407 (2003)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.35 0.4 0.45 0.5 0.55 0.6

Fluid

FCCBCC

T*= kBT/ε = 0.125

packing fraction (η)

scre

enin

g le

ngth

(1/

κσ)

expe

rimen

tal

cond

ition

s

σ = 2.52 µm

T = assume room temperature

ρparticle = assumed 1.196 g/cm3

ε = 3.29 x 10-20 J

1/κ = 0.126 µm

Giving: 1/κσ = 0.05

hard spheres

Simulations at 1/κσ = 0.05, 0.1, and 0.2, all at T* = 0.125

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T vs Packing Fraction Melting Curve for Weitz Experiments

0

0.05

0.1

0.15

0.2

0.25

0.25 0.3 0.35 0.4 0.45

Fluid

FCC StableBCC Metastable

Initial conditions for homogeneous nucleation experiments and simulations

packing fraction (η)

dim

ensi

onle

ss te

mpe

ratu

re (

T*)

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Critical Nuclei: Simulation & Experiment

Time for critical nuclei toappear = 0.03 secs of real time

Total time to end of run = 1 - 3 secsof real time (6 -7 days of computing time)

Nuclei sizes from simulations in range60 – 260 particles

More BCC content in simulated nucleithan experimental nuclei. Simulated nuclei lose BCC content as they grow

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Structure Analysis

• Neither experiments nor simulations give spherical or faceted nuclei

• Neither experiments nor simulations give perfect crystal structures for the nuclei

• Experiments (20 mins of observation) give – BCC 0 – 3%

– FCC 10 – 60%– HCP 20 – 30%– Amorphous 20 – 60% (mostly on exterior)

• Simulations (0.03 secs of observation) give– BCC 10 – 30% (decreases to 0% after further 2 -3 seconds of growth)

– FCC 15 – 45%– HCP 0 – 15%

– Amorphous 30 – 80%

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Effect of Undercooling - Seeded Simulations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

FCC/fluid transition line

FCC/BCC transition line

BCC/fluid transition line

FCC StableBCC Metastable

FCC UnstableBCC Unstable

FCC StableBCC Metastable

FCC UnstableBCC Unstable

FCC MetastableBCC Stable

packing fraction (η)

dim

ensi

onle

ss te

mpe

ratu

re (

T*)

Simulations at 1/κσ = 0.2

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Results

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

FCC/fluid transition line

FCC/BCC transition line

BCC/fluid transition line

BCC seed-BCC crystal

BCC seed-FCC crystal

FCC seed-FCC crystal

FCC StableBCC Metastable

FCC UnstableBCC Unstable

FCC StableBCC Metastable

FCC UnstableBCC Unstable

FCC MetastableBCC Stable

packing fraction (η)

dim

ensi

onle

ss te

mpe

ratu

re (

T*)

“Crystal” means the entire solid box after 1 -3 seconds of real time

12

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.25 0.3 0.35 0.4 0.45

BCC seed - FCC crystal

Fluid

FCC StableBCC Metastable

packing fraction (η)

dim

ensi

onle

ss te

mpe

ratu

re (

T*)

Simulations at 1/κσ = 0.05, if the interactions are very small the metastable seed won’t survive - seeding will not work

Seeded Simulations at WeitzConditions

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Take Home Message About Seeding

• The normal rule is “seed with the desired polymorph in order to grow crystals of that polymorph”

• Based on (limited) simulations, this strategy is not always successful. Success depends on– Strength of interparticle interactions (very weak interactions will

give the stable phase at all undercoolings – seeding with the metastable phase never works)

– For stronger interparticle interactions, success depends on degree of undercooling

- small undercoolings seeding with the metastable phase is successful

- large undercoolings seeding with the metastable phase is NOT successful

– It doesn’t take much deviation from hard spheres for seeding with the metastable phase to be successful at small undercoolings

⇒

⇒

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Stearic Acid

• Seeding technique developed for colloidal particle study • Stearic acid in hexane• Stearic acid has 4 polymorphs (A,B,C,E)

– 6 if include polytypes

• System exhibits a change in stable polymorph – C above 32°C, B below

– Change yields many conditions (temperature and concentration) of interest for polymorph selection

Form C in hexaneGarti, N; Sato,K. eds. Crystallization and Polymorphism of Fats and Fatty Acids. Marcel Dekker, 1988, 273.

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Form B and C Solubility in Hexane

0

2

4

6

8

10

12

15 20 25 30 35 40

BC

Sol

ubili

ty (

g/10

0g s

olve

nt)

T (ºC)Mirmehrabi, M. and S. Rohani. Can. J. Chem. Eng., 2004, 82, 335-342.

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Stearic Acid Form C

c

a

c

b

ab

a= 9.36 Åb= 4.95 Åc= 50.7 Å

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Simulation Details

• Molecular dynamics simulations– NPT ensemble– Gromacs software

• Explicit hexane solvent• United atom model

– CH3, CH2 groups

– Individual atoms in carboxylic acid group

• Current running conditions– 33 ºC, -73 ºC

– 0.073 mole fraction (128 g/L)• Solubility of polymorph C at 47 ºC

– Seeds of various sizes

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Seed Description

• Procedure was developed to equilibrate full crystal of Form C

• Seeds were created from equilibrated crystal

Long Axis View

6 molecules in form of 2 - complete bilayers, 2 - ½ bilayers

Perpendicular view

6 and 12 molecules in other directions

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Changes in Structure

128º

hexane/stearic acid

solution

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Changes in Structure

~90º

Change in angle may indicate “between” polymorph nucleus structure

hexane/stearic acid

solution

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Size of Stable Seeds

• Seeds with ≥ 7-9 layers in the other directions are stable – Seeds with ≤ 6 layers in other directions melt

– Valid for seeds with both 1 and 2 bilayers

• Stable seeds have ≥ 130 stearic acid molecules– ~50 Å in length

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0.75 ns

Steps of Dissolution

0.006 ns1.5 ns

Steps of dissolution follow energy trendMay indicate steps of nucleation

bilayersseparate

stearic acids dissolve from outside in

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Initial Growth

14 layers

Growth in other directions but not in bilayersSupports dissolution observations

Growth may be directional even at small scale

11 layers

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Questions Raised

• What are the steps of growth– Can affect nucleation mechanism

• How does shape affect the minimum seed size?– Shape is not given by Classical Nucleation Theory

• What does structure change indicate?– Possible transition to another polymorph

– Structure is not given by Classical Nucleation Theory

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Acknowledgements

• National Science Foundation Graduate Research Fellowship

• UC Regents Fellowship

• UCSB MRL Central Facilities

• CNSI Computing Facilities, Hewlett-Packard

• Dr. Patricia Soto Becerra

• Doherty Group– Ryan Snyder, Jacob Sizemore, Derek Griffin, Mike Lovette

• Fredrickson Group– Erin Lennon, Won Bo Lee

26Malta V.; G. Celotti; R. Zannetti; A. Martelli. J. Chem. Soc. (B), 1971, 548-553.

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Critical Nuclei by Simulation with Surface Particles Removed to See the Core

Structure

Side view Top view

Nucleus 1

Nucleus 2 Nucleus 3

Nucleus 4