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Classical and Non-classical Nucleation

Mechanisms

Peter VekilovUniversity of Houston

Crystallization is a Miracle!

Crystallization

Nucleation

Crystallization and Nucleation

…

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Nucleation Kinetics Is Different

Josiah Willard Gibbs (1839 - 1903)

… and this has been strictly shown for Gibbs only

Only God and Gibbs never erred

E.D. Shchukin

Free Surface Energy Surface Free Energy

, , , ,

–

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Molecular Origins of Surface Free Energy

l

G

rl

Excess G at liquid side: enthalpicExcess G at vapor side: entropic

Molecular Origins of Surface Free Energy

c

z

G

z

Liquid Gas

Liquid Gas

l +g

– hatched area

Position of interface: Zero concentrationEqual energy

The contact angle of a droplet with a solid

r

2 1

1

13 2 3

Young equation

If the liquid and substrate are similar , ≪ , cos 1 and a 0

If the liquid is repelled by the substrate , cos 0 and /2.

The pressure within a small gas bubble

= xy – (x + dx)(y + dy)  –(xdy + ydx)

G = Vp + .

At equilibrium, G = 0 and 

.

2Young‐Laplace relation

and

2Gibbs‐Thomson relation

= p Vm

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Surface Free Energy of Crystals

81 atoms 36 broken bonds

81 atoms 44 broken bonds

Surface Free Energy of Crystals

2,

2

The Thurnbull Rule

0.3Δ Ω /

2

2Δ

Ω /

Classical Nucleation Theory: Thermodynamics

n molecules from solution into a crystalline cluster

Gibbs JWOn the equilibrium of heterogeneous substances

Trans. Connect. Acad. Sci. (1876) 3, 108;(1978) 16, 343

Solution—supersaturated: soln > crystal,   = soln – crystal > 0

C2

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Classical Nucleation Theory: Thermodynamics

Free energy gain = ‐ n

Free energy loss = 6n2/3creation of new surface

G(n) = Δ 6 Ω /

Δ4/

∗ 64 ΩΔ

Δ ∗32 ΩΔ

n2/3G

G*n

n*

G(n)

‐ n

Classical Nucleation Theory: Thermodynamics

∗ 64 ΩΔ

Δ ∗32 ΩΔ

12

∗Δ

n2/3G

G*n

n*

G(n)

‐ n

G

n

G*

G≠

Creation of Chemical bonds in the Crystal The Structure of the Crystal Nucleus

Suggested nuclei shapes

Milchev, Contemp. Phys. 32 (1991) 321

0 4 8 12 200

4

8

12

n*N

ucle

atio

n W

ork

a

b

c

d

eg f

16

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Radius of 2D nucleus

ΩαΔμ

∆

Critical Radius

Supersaturation

21

Experimental Tests Thermodynamics of Hematin Crystallization

0 20 40

0.15

0.30

0.45

BulkAFM

c e(m

M)

T (oC)

1/T (1/K)

lnc e

0.0032 0.0036-3-2-1

⁄∆

The crystallization enthalpy ∆ = -37±8 kJ mol-1

22

Radius of 2D nucleus

ΩαΔμ

α 0.3Δ Ω /

Critical Radius

Turnbull’s Rule

= 27 4 mJ m-2

23

Experimental Tests

∆ = -37±8 kJ mol-1

Kinetics of Crystal Nucleation

Nucleation rate J: nuclei per unit volume per unit time

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Max Volmer(1885 - 1965)

Classical Nucleation Theory: Kinetics

Volmer M (1939) Kinetik der Phasenbildung (Steinkopff, Dresden)

J

C

exp∆ ∗

Assumes that crystal nuclei form directly in solution

Predicts steep

Jo = ?????

, , , ,

The General Aggregation Scheme

∂ ,∂ 1, 1, ,

∂ 1,∂ 1, 2, , ,

, , 1,

The Szilard Assumption

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The “Equilibrium” Concentration of Nuclei

Y. B. Zel'dovich, Acta Physicochimica URSS 18, 1 (1943).

Diffusion in the Space of n over a Barrier The Zeldovich Factor

∗ ∗

z = ∗

∗ /

z 0.1 – 0.01

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The Nucleation Rate Law

∗ ∗

∗ exp ∆ ∗⁄

∗ NOT limited by diffusion

- Experimentally measured- Evaluated based on reaction kinetics

G

n

G*

G≠

Time Evolution of Nucleation

2

3 2 ∗

=  1/JstV + d/R

Non-classical Behaviors

Solution-to-crystal spinodal: n* ≤ 1

Two-step nucleation of crystals

Non-equilibrium crystal shape

Mysteries of Nucleation Kinetics

5 10 15 20 25 300

0.2

0.4

TL-L

TL-L

Clys= 50 mg/mlClys = 80 mg/ml

Temperature T [°C]

Hom

ogen

eous

Nuc

leat

ion

Rate

J[c

m-3

s-1]

Non-monotonic behavior of J(T)

O. Galkin & P. G. Vekilov,PNAS 97, 6277 (2000)

Nucleation rate lower by 109then prediction of CNT

J0(experiment) ~ 1010 cm-3s-1J0(CNT) ~ 1019 cm-3s-1

J0(CNT) = f+zC

f+ 300 s-1z 0.1C 1018 cm-3

)exp(*

0 TkGJJB

Sharp leveling of J()

Large data scatter

O. Galkin et al., JACS 122, 156 (2000)

Vekilov PG (2010) Nucleation. Crystal Growth & Design 10(12):5007-5019.

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Determination of Nucleus Size

Nucleation theoremD. Kashchiev, D. Oxtoby JCP 100 (1994) 7665

)1()/(

)(lnB

* OTkd

Jdn

allows determination of nuclei sizes (10 4 1)

O. Galkin, P.G. Vekilov, JACS 122 (2000) 156

..

0

0.2

0.4

2.5%

2.5 3.0 3.50.01

0.1

1

3%

4%

2.5% 3% 4%

L-L coexistence

L-L coexistence

Hom

ogen

eous

Nuc

leat

ion

Rat

e J

[cm

-3s-

1 ]

Supersaturation = In(C/Ceq)

10.0oC12.6oC15.0oC

2.8 3.2 3.610-2

10-1

100

Nucl

eatio

n Ra

te J

[cm

-3s-

1 ]

Supersaturation / kBT

n* = 1

• Spinodal can be defined from n* 1

Solution-to-crystal Spinodal

L.F. Filobelo, et al., J. Chem. Phys. 123, 014904 (2005)

At and below spinodal G* < kBT

Why is the maximum in J(T) sharp?

0 100 200 300

-10

0

10

20

30

40

50

Tem

pera

ture

[oC

]

Concentration [mg/ml]

Liquidus

Liquid-liquid (L-L) spinodal

L-L coexistence

Gelation line

J(T) reaches sharp max at solution-crystal spinodal

J() levels off at solution-crystal spinodal

Direct and Two-step Nucleation

Concentration

Stru

ctur

e

…

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The Two-step Nucleation Mechanism

1 2

Tem

pera

ture

Protein Concentration

Solubility

Gelation

Binodal

Spinodal

The Two-step Nucleation Mechanism

What are the structuresthat serve as precursors above the L-L coexistence line and persist in the homogeneous region?

?

1 2

Protein-Rich ClustersAtomic force microscopy,lumazine synthase

O. Gliko, et al., JACS 127, 3433 (2005)

0

100

200

0 2.5 5 7.5 10Surface Coordinate [m]

75 nm

10-4 10-2 100 102 1040.0

0.5

1.0

Delay Time [ms]

G()

Dynamic light scattering,lysozyme

W. Pan, et al., Biophys. J. 92, 267 (2007)

10 1000

5

10

Exp. 1 Exp. 2 Exp. 3

Cluster Radius R2 [nm]

Con

cent

ratio

n n 2

[107

cm

-3]

Reproduces results on R2 and n2 from dynamic light scattering

Brownian microscopy

Y. Li, et al., Rev. Sci. Instr. 82, 053106 (2011)

Do Crystals Nucleate in the Clusters?You can observe a lot by watching.

Yogi Berra

Depolarized Oblique Illumination Microscopy

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Do Crystals Nucleate in the Clusters?Depolarized Oblique Illumination Microscopy

Phenomenological Theory of Two-step Nucleation

0 1 2

– mean first-passage time

J = -1 , 2 – rate-limiting

)(1

)()()(

)(1

220

1

0 TuTuTuTu

Tu

TkG

UUTC

TkGTCk

J

B

liquid

B0

0

11

*2

12

exp1),(

)exp(

TkECkC B/expexp1 110

2

2

21)(

**2

spe

e

e TTTT

TTETG

Single adjustable k2 reproduces 3 complex kinetic curves

W. Pan, et al., J. Chem. Phys. 122, 174905 (2005)

276 280 284 288 292 296 3000.0

0.1

0.2

0.3

0.4

0.5

Nuc

leat

ion

Rate

J [c

m-3s-

1 ]

Temperature T [K]

80 mg ml-1

50 mg ml-1

20 30 40 50 600.0

0.1

0.2

Nucl

eatio

n Ra

te J

[cm

-3s-

1 ]

Concentration C [mg ml-1]

T = 12.6 oC

Nucleation barrier on approach to spinodal

Viscosity inside dense liquid

TkG

UUTC

TkETCk

J

B

B

exp1),(

)exp(

0

11

212

The Pre-exponential Factor in the Nucleation Rate Law

)exp(*

0 TkGJJB

From experiments: J0 ~ 1010 cm-3s-1

From classical theoryJ0 ~ 1018 cm-3s-1

From phenomenological theory:

fractionvolumecluster,1)exp(0

1

TkG

UU

B

0 50 100 150 200 250

10-7

Volu

me

Frac

tion

Time of Monitoring [min]

Low J0—due to nucleation within clusters

inside clusters 10 cP

-1 108

Explains low Jo0 1 2 3

0.00

0.04

0.08

0.12

0.16

2.2 2.4 2.6 2.8 3.0 3.210-2

10-1

/ kBT

Nuc

leat

ion

Rat

e [c

m-3 s

-1]

Concentration [mg ml-1]

Why is the Two-step Mechanism Faster?

Vekilov, P. G. Crystal Growth & Design 2010, 10, 5007

exp∆ ∗

∆ ∗ 10-20 J

∆ ∗ 32

0.2 mJ/m2

10-1

10-22.4 2.8 3.2

/kBT

Nucleation rate data for insulin

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94 frames Size: 9.5 x 9.5 m250 s per frameReal time: 95 min

Growth of Insulin Crystals

I. Reviakine, et al., J. Am. Chem. Soc.125, 11684 (2003)O. Gliko, et al., Phys. Rev. Lett 90, 225503 (2003)

Estimate of Crystal Solution Surface Free Energy

Lc 320 nmΩΔ

28 mJ/m2

Nucleation in dense liquid clusters lowers by ~ 140× and G* by 106Increases J by 14 orders of magnitude

Reviakine, I., et al., J. Am. Chem. Soc. 125, 11684-11693, (2003)

The Two-step Mechanism for Solution Crystallization

Protein CrystallizationO. Galkin & P. G. Vekilov, PNAS 97, 6277 (2000)P. G. Vekilov, Cryst. Growth & Design 4, 671 (2004)M. Sleutel, A. van Driesche, PNAS (2013)

Glycine, ureaB. Garetz, et al., Phys. Rev. Lett. 89, 175501 (2002)J.E. Aber, et al., Phys. Rev. Lett. 94, 145503 (2005)D.W. Oxtoby, Nature 420, 277 (2002)

Colloid crystalsM. E. Leunissen, et al., Nature 437, 235 (2005)J. R. Savage and A. D. Dinsmore, PRL 102, 198302 (2009)T. H. Zhang and X. Y. Liu, JPCB 111, 14001 (2007)

NaClO3R.Y. Qian, G.D. Botsaris, Chem. Eng. Sci. 59, 2841 (2004)

Calcite nucleation in bulk solutionsL. Gower, Chem. Rev. 108, 4551 (2008)H. Coelfen, et al., Science 322, 1819 (2008)E. M. Pouget, et al., Science 323, 1455 (2009)A.F. Wallace, et al., Science 341, 885 (2013)

+

1 2

Nanohorn with hook

Dense liquid droplet

Crystal nucleus

Largecrystal

=

Br

Br

Br

The Case of Y-Br

Harano, K. et al. Nature Mater. 11, 877–881 (2012).

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HbS Polymer Nucleation

D(T) much stronger than R(T)contradicts 1-step nucleation and agrees with 2-step

Polymers are perpendicular to plane of polarization of polarized light

Dependencies of Vl of mesoscopic metastable clusters on C and T follow those of nucleated polymers

0

30

60

90

120

150

180

210

240 300

330

Time of Monitoring [min]

0 20 40 60

200

400

600

800

10oC 15oC 20oC 25oC 30oC

Rad

ius

[nm

]

Clusters are precursors for polymer nucleiO. Galkin, P.G. Vekilov, J. Mol. Biol. 336, 43 (2004)

O. Galkin, et al., J. Mol. Biol. 365 425 (2007)O. Galkin, et al., Biophys. J. 92, 902 (2007)

P.G. Vekilov, Brit. J. Haematol. 139, 173 (2007)

0 5 10 15 20 250.0

0.2

0.4

0.6

D [s]

1/R

[s

m-1

]

Fe

O-

O-

O

ON

N-

N-

N

CH3

CH3

CH3

CH3CH2

CH2

H H

H H

(a)

(b)

So What: the Heme and Sickle Cell Polymerization

0 30 60 90 12010-9

10-7

10-5

10-3

67 mg ml-1 96 121 121, 66 M 131 178

Volu

me

Frac

tion

Time t [min]

Dependence of Cluster Volume Fraction on HbS Concentration

2 for hemoglobin S

Increases with C from 67 to 96 mg ml-1

Similar at 96, 121, 131, and 178 mg ml-1

as per theory

Heme addition: 2 higher by 80×

The nucleation rate ??

60 80 100 120 140 160 18010-6

10-5

10-4

10-3

Concentration C [mg ml-1]

Ava

rage

Vol

ume

Frac

tion

107

108

10

100

5 10 15 20 25 30 350.1

1

10

Nuc

leat

ion

Rat

e J

[cm

-3s-

1 ]D

elay

Tim

e

[s]

290 mg ml-1, 0.16 mM271, 0.26 267, 0.28

280 mg ml-1;265; 245

230 mg ml-1; 220; 210; 201

Temperature T [oC]

Gro

wth

Rat

eR

[ m

s-1]

The Nucleation Rate

In the presence of heme, J is ~ 100× faster

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How Can We Image the Critical Clusters?1)  They exist for a very short time2)  They are freely floating in the solution

• Protein crystallization is very slowAt = 1.2 kBTSolution/crystal net exchange frequency f = 0.065 s-1 = 1 molecule/50 s/active site

At dG/dn = 0 – even slowerComparable to AFM imaging time

Dcluster =D1

nedge=

3 10-7 cm2/s5 = 6 10

-8 cm2/s

From x2 = 2D = (100 m)2/ 2Dcluster = 850 s ≈ 15 min

• A cluster needs 15 min to reach surface after nucleation 100 m above substrate• Ferritin molecules readily adsorb on glass

Near-critical and sub-critical clusters could be viewed by AFM

0.2 m

• Clusters are Brownian particles

Adsorption of Ferritin Molecules on the Substrate

Full coverage with a disordered monomolecular layer of ferritin – 2D ferritin glass

50 nm

10 nm

Strong adsorption, no surface mobility of the molecules No crystallization occurs in the adsorbed layer

All crystalline aggregates seen on the surface come from the solution bulk and lie on top of adsorbed layer

Clusters Are Formed in the Bulk and Land on the Surface

t = 0

50 nm

t = 430 s

For ~7 min the monitored cluster looses 2 molecules Another cluster lands next to it from the solution  Observed for many clusters, supported by indirect evidence

Structures of Aggregates Adsorbed on Substrate

50 nm

(a) (b)

(c) (d)

Aggregates containing 2, 4, 6, ~16 and ~20 molecules

Molecules in the 6-membered aggregate in (b) are arranged in 2 rods with 4 and 2 molecules

Molecules in larger aggregates (c) and (d) form a single layerconsisting of a few rods of 4-5 molecules

Disordered aggregates also seen

Ordered aggregates are in dynamic exchange with solution, disordered aggregates are “dead”

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Structures of Aggregates Adsorbed on a Crystal

Better visibility of cluster structure, likely due to stronger adsorption on structured surface of crystal

100 nm

896 s

• Arrangement of molecules – same as in the bulk of a large crystal, intermolecular distance = 13 nm

• Shape: 110 rods forming 1 or 2 (110) layers• Structure compatible with clusters seen on the substrate

S.-T. Yau, P.G. Vekilov, Nature 406 (2000) 494.

100 nm

0 s

Dynamics of Exchange with Solution

Blue molecules: detach from cluster after frame recorded

Red molecules: attach to cluster after previous frame recorded

Clusters are in dynamic exchange with solution: these are not dead-end aggregates

Net attachment frequency: 4 molecules / 43 s 0.1 attachment

event / second 20 possible sites f = 0.005 s-1 per site

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t = 0

50 nm

t = 384 s

t = 640 s t = 1102 s

Random Selection of Cluster Behavior

Larger cluster dissolves over long times, smaller cluster adds and loses 1 molecule

Seeming violation of Oswald ripeningrules?

Oswald ripening describes evolution of large populations of crystallites

Are These Precrystalline Clusters?Do they evolve into crystals?

0 s100 nm 170 s 340 s

0

25

-250 250 500

Surface Coordinate [nm]

Heigh

t[nm

]

• (111)-faceted microcrystal consisting of 3 layers formed in solution:

- inclined with respect to underlying face- misfit upon incorporation by oncoming step

• Contains ~ 140 molecules: slightly larger than the near critical clusters

• Slowly grows• It has evolved from near-critical clusters similar to those above

Do These Clusters Belong to the Crystal Nucleation Pathway?We have shown that the clusters:

Form in the solution bulk

Their structure is representative of the structure of the crystal nuclei

100 nm

100 nm

385 s100 nm

0 s100 nm

Evolve into crystals

Size is inversely proportional to supersaturation

Grow or dissolve over longer time periods

The net exchange frequency is very low, indicative of near equilibrium

Are in dynamic exchange with the medium

Have arrangement of molecules same as in crystal

These are subcritical and near critical clusters in the nucleation pathway of ferritin crystals

S ingle molecules rods planar nuclei crystals

V

IV

IIIIII

• All 5 steps are evidenced by the images above (110) layers stack to form a crystal faceted by (111) planes

Suggested Nucleation Pathway

S.‐T. Yau, P.G. Vekilov, Nature 406 (2000) 494

Rather thanCompact structure of all subcritical clusters

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Shape of Colloid Crystal Nuclei

U. Gasser, E. Weeks, A. Schofield, P. Pusey, and D. Weitz, Science 292, 258 (2001).

Why Would Spherical Molecules Form Linear and Planar Clusters?• If long-range repulsion acts, after a dimer is formed, the axis sites are

preferred loci for attachment of third molecule, etc.A.J. Hurd, D.W. Shaefer PRL 54 (1985) 1043

U

r

Long-rangerepulsive potential

+ +

Do All Crystals Follow the Two-step Mechanism?Ferritin: nucleus size fully explained by classical theory

S.‐T. Yau, P.G. Vekilov, Nature 406 (2000) 494

50 nm

= 1.6 kBTn* = 4 moleculesn* scales ~ with 1/, as per Gibbs-Thomson relation

R. Kaischew and I. Stranski, Z. Phys. Chem. B35 (1937) 427

— intermolecular bond energyS.‐T. Yau et al., PRL 85 (2000) 353

n* =

= 3.2 kBT

at  = 1.6 kBT n* = 4

2