Kinetics and Thermodynamics of Multistep Nucleation and Self-assembly in Nanosize Materials Brussels, March 25-26, 2010 The Two-step Mechanism of Nucleation of Crystals from Solution Peter G. Vekilov, Oleg Galkin, Luis Filobelo, Weichun Pan, Anatoly Kolomeisky (Rice), Dimo Kashchiev (IPC), Vas Lubchenko Department of Chemical and Biomolecular Engineering, Department of Chemistry, University of Houston NIH, NASA, NSF, Welch Foundation
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Kinetics and Thermodynamics of Multistep Nucleation and Self-assembly in Nanosize Materials Brussels, March 25-26, 2010 The Two-step Mechanism of Nucleation.
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Kinetics and Thermodynamics of Multistep Nucleation and Self-assembly in Nanosize Materials
Brussels, March 25-26, 2010
The Two-step Mechanism of Nucleation of Crystals from Solution
Peter G. Vekilov, Oleg Galkin, Luis Filobelo, Weichun Pan, Anatoly Kolomeisky (Rice), Dimo Kashchiev (IPC), Vas Lubchenko Department of Chemical and Biomolecular Engineering,Department of Chemistry, University of Houston
NIH, NASA, NSF, Welch Foundation
The Goal: Crystals with “Just-right”…
Number PolymorphMorphologyHabitSizeSize distribution
Requires data on:
Solution PChem Phase
diagrams Metastable
statesNucleation
mechanismsGrowth
mechanismsAgglomeration…
Crystallization
Nucleation
Crystallization and Nucleation
Crystallization
Nucleation
Crystallization and Nucleation
Crystallization as Sequential Transition along Two Order Parameters
Concentration
Str
uct
ure
…
Classical viewpoint: direct nucleation along a “diagonal line” envisioned;
Protein Concentration
Tem
per
atu
re
Solubility
Gelation
Binodal
Spinodal
Two-step mechanism: suggested by t W & F, T & O for critical point for L-L phase separation for proteins
Everywhere else in phase diagram—classical crystal nucleation predicted P.R. ten Wolde, D. Frenkel,Science 277 (1997) 1975
V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 109 (1998) 223
The Two-step MechanismConcentration
Str
uct
ure
…
It operates in all areas of the phase diagram
It may apply to all crystals (and other ordered solids) forming in solution
Protein Concentration
Tem
per
atu
re
Solubility
Gelation
Binodal
Spinodal
Galkin, O. & Vekilov, P. G. (2000) Proc. Natl. Acad. Sci. USA 97, 6277
Vekilov, P. G. (2004) Crystal Growth and Design 4, 671
5 10 15 20 25 300
0.2
0.4
TL-L
TL-L
Clys = 50 mg/ml Clys = 80 mg/ml
Temperature T [°C]Hom
og
en
eou
s N
ucle
ati
on
Rate
J [
cm-3 s
-1]
The Nucleation Rate
Maximum in J(T) Exponential increase at intermediate DT’s; by weak decrease at
higher DT’s T of maximum shifts with concentration
Protein Concentration
Tem
per
atu
re
Solubility
Gelation
Binodal
Spinodal
Two steps:Which One is Rate Determining
Critical level of ordering
Critical size of ordered nucleus
Concentration
Str
uct
ure
…
*2G
*2G
*1G
*1G
s
olut
ion
de
nse
liqui
d
cr
ysta
ls
DGL-L
DGL-L
Fre
e E
ner
gy
G
Nucleation Reaction Coordinate
Rate of cluster formationJ1 ~ J01exp(– /kBT)
Rate of nucleation within clustersJ2 ~ J02exp(– / kBT)
Is > ?
Is J01 more important than ?
Is J02 more important than ?
*1G
*2G
*1G *
2G
*1G
*2G
Nucleation of Dense Liquid Droplets
3.53 s 10.58 s6.09 s
0.96 s 1.92 s 7.37 s
T – TL-L = 0.7 oC
T – TL-L = 1.3 oC
• Number of droplets increases with time• Faster nucleation at higher DT’s Characteristics of nucleation regime of droplet generation
Nucleation Rate of Dense Phase Droplets
Time [s]
Dro
ple
ts in
Vie
wfi
eld
0 2 4 6 8 100
100
200
300
400
Nucleation Rate = 4.3 x 109 cm-3s-1
• Number of droplets increases in time—nucleation regime
• Nucleation rate ~109 cm-
3s-1
significantly higher than rates of crystal nucleation ~ 0.1 – 1 cm-
3s-1
M. Shah, et al., J. Chem. Phys. 121 (2004) 7505
Structuring of dense liquid quasi-droplet is the rate determining stage Equilibrium between solution and clusters: msolution = mclusters Dm(solution,crystal) = Dm(clusters,crystal)
10.0oC12.6oC15.0oC
2.8 3.2 3.610-2
10-1
100
Nuc
leat
ion
Rat
e J
[cm
-3s-1
]Supersaturation / k
BT
n* = 1
• Spinodal can be defined from n* 1
• Spinodal – boundary between metastable and unstable two – phase areas
Other proteins: • Ferritin crystals grown at
s = 4.2, where n* 1 • Are protein crystals always
grown in spinodal regimes?
L.F. Filobelo, et al., J. Chem. Phys. 123, 014904 (2005)
Why is the maximum in J(T) sharp?
20 mm
0 100 200 300
-10
0
10
20
30
40
50
Tem
per
atu
re [o C
]
Concentration [mg/ml]
Liquidus or solubility of crystals
L-L spinodal
L-L coexistenceSolution-crystalspinodal
Gelation line
Pre-exponential Factors and Barriers for Structuring
D.N. Petsev, et al., J. Phys. Chem. B 107 (2003) 3921
L.F. Filobelo, et al., J. Chem. Phys. 123 (2005) 014904
Why is the maximum in J(T) sharp?
0 100 200 300
-10
0
10
20
30
40
50
Tem
per
atu
re [o C
]
Concentration [mg/ml]
Liquidus
Liquid-liquid (L-L) spinodal
L-L coexistence
Gelation line
J(T) reaches sharp max at solution-crystal spinodal
Phenomenological Theory of Two-step Nucleation
0 1 2
t – mean first-passage time
J = t-1 , 2 – rate-limiting
)(
1
)()(
)(
)(
1
220
1
0 TuTuTu
Tu
Tu
)exp(1
)exp(
0
1
22
TkG
UU
TkE
U
J
B
B
),( 1
122 TC
TCkU
TkECkC B/expexp1 110
2
2
21)(
*
2spe
e
e TT
TT
TT
ETE
276 280 284 288 292 296 3000.0
0.1
0.2
0.3
0.4
0.5
Nu
clea
tio
n R
ate
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
Nu
clea
tio
n R
ate
J [c
m-3s-1
]
Concentration C [mg ml-1]
Single adjustable k2 reproduces 3 complex kinetic curves
W. Pan, et al., J. Chem. Phys. 122, 174905 (2005)
T = 12.6 oC
Nucleation barrier on approach to spinodal
Viscosity inside dense liquid
The Pre-exponential Factor in the Nucleation Rate Law
R.P. Sear, J. Phys. Chem. B 110 (2006) 21944
)exp(1
)exp(
0
1
22
TkG
UU
TkE
U
J
B
B
)exp(*
0 Tk
GJJ
B
From experiments: J0 ~ 1010 cm-3s-1
From classical theoryJ0 ~ 1020
cm-3s-1
???
From phenomenological theory:
fractionvolumecluster,1
)exp(0
1
Tk
G
U
U
B
0 50 100 150 200 250
10-7
Vo
lum
e F
rac
tio
n
Time of Monitoring [min]Low J0—due to nucleation within clusters
The Two-step Mechanism for Other Crystals
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)
Charged colloid crystalsM. E. Leunissen, et al., Nature 437, 235