Multi-site Models to Accurately Determine the Distribution of Kink Sites adjacent to Low Energy Edges

Michael A. Lovette, Michael F. Doherty

Department of Chemical Engineering, University of California Santa Barbara

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• Crystal shape impacts– Downstream processing efficiency (filtering, flowability, tableting)– End use functionality (catalytic activity, bio-availability)

• The distribution of kink sites is the key anisotropic parameter that determines crystal shapes when grown from solution– An accurate prediction of the distribution of kink sites is required to predict

shapes from first principles using mechanistic growth models

• The classic models for determining kink site distributions were developed assuming low kink densities and strong intermolecular interactions– Frequently not the case for solution growth of small organic molecules0.00

0.25

0.50

0.75

1.00

0 1 2 3 4 5 6 7 8

P

w (kbT)-1

PK - oldPK - new

Motivation

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Kink Sites• Unoccupied sites occurring adjacent to a disturbance along an edge • Play an essential role in crystal growth and dissolution • Two unique properties that allow for growth in a “self-sustaining” manner

1. Maintain the free energy of the edge constant (thermodynamic)2. Create another kink site (renewability)

Figure 3. Kink sites on the (100) face of insulin at low supersaturation. Consistent with the

classical view of layered growth.

(left) Georgiou and Vekilov, PNAS, 2006.

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• Today’s limited focus: – Disturbances arising from thermal roughening of the edge– Edge’s with their most probable configuration of disturbances and disturbance free sites

– Resulting average distance between successive disturbances

Disturbances

Figure 4. Conventional thought process for determining the most probable (native)configuration of the edge with disturbances caused by thermal roughening.

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Figure 5. Kink sites on the (100) face of insulin at low supersaturation. Consistent with the

classical view of layered growth.

Kink Sites and Disturbances• Classically (Frenkel, J Phys USSR, 1945) assumed a 1:1 correspondence • Kinks have two unique properties

1. Maintain the free energy of the edge constant (thermodynamic)2. Create another kink site (renewability)

• Geometric minimum average spacing of (Vekilov, Cryst Growth Des, 2007)– but

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Kink Sites and Disturbances• Classically (Frenkel, J Phys USSR, 1945) assumed a 1:1 correspondence • Kinks have two unique properties

1. Maintain the free energy of the edge constant (thermodynamic)2. Create another kink site (renewability)

• Geometric minimum average spacing of (Vekilov, Cryst Growth Des, 2007)– but

Figure 6. Edge segment on the (111) face of apoferritin. Inconsistent with the 1:1

correspondence between kinks and disturbances.

(left) Yau et al, J Mol Biol, 2000.

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Models for Site Distributions• Single site model (1:1 correspondence)

– Kink site unoccupied site adjacent to a disturbance– probability of a single event (a disturbance) occurring

• Multi-site models– Kink site site obeying both properties of a kink– is based on the conjoined probability of 4 independent events occurring

Types of unoccupied sites (Kossel (100) face, KC)

Disturbance Single Site Model Multi-Site Model

Yes Kink Site

Thermodynamic Kink

(obeys Prop. 1)

Kink Site (obeys Prop. 2)

Ledge

Double VacancyEdge Vacancy (Hole)

No Edge Adatom Edge Adatom

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Multi-Site Models• 4 independent events each with 3 possible outcomes

1. Negative disturbance:2. Positive disturbance:3. No disturbance:

• 34 = 81 different possible states of the edge segment• 81 states 9 families for KC

– KC is universally applicable for systems with 2 centrosymmetric PBCs• 81 states 11 families for FCC

– FCC case is universally applicable for 3 centrosymmetric PBCs

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(c)

x

Model Development: KC

(0,0) (1,0)

(0,1)

(-1,0)

(0,-1)

(a)x

State Mirror P(a) Yes P0P0P0P0

(b) No 2P0P-P0P0

Figure 7. Disturbance free edge segment (a), and edge segment with a single disturbance (b) that results in the formation of an adjacent kink site. (c) is the “mirror” image of (b).

(b)

x

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Grouping States into Families(b)

x

(c)

x

(d)

x

(e)

x

State Mirror P(b) No 2P0P-P0P0

(c) No 2P-P-P0P0

(d) No 2P+P-P0P0

(e) No 2(P-+P0+P+)P-P0P0

Figure 8. Family of states corresponding to a conventional kink site. The filled green circles in (e) are hereafter “or” statements.

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Kinks: KC

Figure 9. Two families corresponding to Kink sites at the position marked “x”. Note the yellow site which becomes a kink after an incorporation event into the grey site.

(a)

x

(b)

x

(c)

x

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Thermodynamic Kinks: KC

Figure 10. Families corresponding to thermodynamic kink sites. (c) is a kink, (d) is a ledge, and (e) is a double vacancy. The renewability property is not valid in (d) and (e).

(d)

x

(e)

x

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Thermodynamic Kinks: KC

State Mirror P P (bw = 0)

(a) Kink No 2 P-P0 (P0+P-) 12/81

(b) Kink No 2 P-P-P- 6/81

(c) Kink No 2 P-P-P0P+ 2/81

(d) Ledge No 2 (P0+P+)P-P- 12/81

(e) DV No 2 (P0+P+) P-P0P+ 4/81

All TKs No ------ 36/81 = 0.44

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Not Kinks: KC(b)

x

(d)

x

(c)

x

(a)

xFigure 11. States of the edge

segment that do not obey the thermodynamic equivalence

property of a kink site. (a) is an edge vacancy, (b-d) are edge

adatoms.

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Not Kinks: KC

State Mirror P P (bw = 0)

(a) EV Yes P-P+ 9/81

(b) AA Yes P0P0 9/81

(c) AA Yes P+P- 9/81

(d) AA No 2 P0P- 18/81

All NKs ----- ------ 45/81 = 0.56

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Probabilities: KC

0.00

0.25

0.50

0.75

1.00

0 1 2 3 4 5 6 7 8

P

w (kbT)-1

Kink

Edge VacancyEdge Adatom

DisturbanceThermodynamic Kink

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Justifying ntk

(a)

7 7 33 ntk = 5nk

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Justifying ntk

(b)

5 5 55 ntk = 5nk

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Justifying ntk

(c)

1 1 99 ntk = 5nk

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Model Development: FCC case(a)

(0,0) (1,0)(-1,0)

(0,-1)(-1,-1)

(0,1)(-1,1)

x

(b)

x

(c)

+-

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Thermodynamic Kinks: FCC

(d)

x

(b)

x

(a)

x

(c)

x

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Thermodynamic Kinks: FCC(e)

x

(f)

x

(g) (h)

x x

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Not Kinks: FCC(a)

(b)

x

x(c)

xHalf adatom site (game changer)

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Probabilities: FCC

0.00

0.25

0.50

0.75

1.00

0 1 2 3 4 5 6 7 8

P

w (kbT)-1

Kink

Edge VacancyEdge and Half Adatom

DisturbanceThermodynamic Kinks

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Comparing nN

o. o

f Mol

ecul

es

0

5

10

15

0 1 2 3w (kbT)-1

nk (FCC)ntk

nk (Kossel)ntk

l/a

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Conclusions and Future Work• Multi-site models give accurate kink site distributions for

edges with w < 3 kbT – Merit implementation into shape prediction modeling efforts and

experimental studies

• These models are universally applicable for centrosymmetric systems – The development and application of these models for non-

centrosymmetric systems is a potential avenue for future research

• These distributions are based on an equilibrium assumption– The rate into and out of the equilibrium state should be determined